answer：Alright, so I have this problem here: "The negation of the proposition '∃x∈Z, x²=2x' is ______." Okay, let's break this down step by step. First, I need to understand what the original proposition is saying. It says, "There exists an integer x such that x squared equals 2 times x." So, in simpler terms, it's saying that somewhere in the set of all integers, there's at least one number that when you square it, it's the same as doubling that number.Now, I need to find the negation of this proposition. Negation, in logic, means the opposite. So, if the original statement is saying that there exists such an x, the negation should be saying that there does not exist such an x. But how do I express that formally?I remember that in logic, the negation of an existential quantifier (∃) is a universal quantifier (∀) with the negated statement. So, if the original statement is ∃x P(x), the negation would be ∀x ¬P(x). Applying that here, the negation of "There exists an integer x such that x²=2x" should be "For all integers x, x² does not equal 2x."Let me write that out to make sure it makes sense. The original proposition is ∃x∈Z, x²=2x. The negation would be ∀x∈Z, x²≠2x. That seems right. But wait, maybe I should check if there actually exists such an integer x to see if the original statement is true or not, just to understand it better.So, let's solve the equation x²=2x. If I rearrange it, I get x² - 2x = 0, which factors to x(x - 2) = 0. So, the solutions are x=0 and x=2. Both of these are integers, which means the original statement is true because there do exist integers (specifically 0 and 2) that satisfy the equation.But since the problem is asking for the negation, and we've established that the original statement is true, that means the negation should be false. So, the negation is "For all integers x, x² does not equal 2x," which is false because we know that x=0 and x=2 do satisfy the equation.Just to make sure I'm not missing anything, let's think about what the negation really means. It's saying that no integer x satisfies the equation x²=2x. But we've already found that x=0 and x=2 do satisfy it, so the negation can't be true. Therefore, my negation seems correct.I also recall that in formal logic, the negation of ∃x P(x) is indeed ∀x ¬P(x). So, applying that rule here, I'm confident that the negation is correctly formed.Another way to think about it is to consider the truth values. If the original statement is true, its negation must be false, and vice versa. Since the original statement is true (because x=0 and x=2 are solutions), the negation must be false. And that's consistent with what I found earlier.So, putting it all together, the negation of "There exists an integer x such that x squared equals 2x" is "For all integers x, x squared does not equal 2x." This makes sense logically and aligns with the rules of negating quantifiers in formal logic.I think I've covered all the bases here. I've broken down the original proposition, applied the rules of negation, checked the solutions to confirm the truth values, and verified that the negation aligns with logical principles. I feel confident that this is the correct negation.

answer：First, I need to determine the new length of the fishing rod after increasing its original length by 1.33 times.The original length of the fishing rod is 48 centimeters.To find the extended length, I will multiply the original length by the increase factor:48 cm × 1.33.Calculating this gives:48 × 1.33 = 63.84 cm.Therefore, the length of the extended fishing rod is 63.84 centimeters.

answer：Alright, so I have this geometry problem about a tetrahedron (ABCD). I need to draw the bisector plane of the dihedral angle around edge (CD). This plane intersects edge (AB) at a point (E), and I have to show that the ratio (frac{AE}{BE}) is equal to the ratio of the areas of triangles (ACD) and (BCD), that is, (frac{t_{ACDDelta}}{t_{BCDDelta}}).First, I need to recall what a dihedral angle is. A dihedral angle is the angle between two planes. In the context of a tetrahedron, the dihedral angle around edge (CD) would be the angle between the two faces that meet at edge (CD). These faces are triangles (ACD) and (BCD). So, the bisector plane of this dihedral angle would be a plane that splits this angle into two equal parts.Now, this bisector plane intersects edge (AB) at point (E). So, point (E) is somewhere along edge (AB), and I need to find the ratio of the lengths (AE) to (BE).I think I should start by visualizing the tetrahedron. Let me sketch it out mentally: tetrahedron (ABCD) with vertices (A), (B), (C), and (D). Edges (AC), (AD), (BC), and (BD) connect to form triangles (ACD) and (BCD) around edge (CD). The dihedral angle between these two triangles is being bisected by a plane, and this plane intersects edge (AB) at point (E).I remember that in plane geometry, the angle bisector theorem relates the ratio of the adjacent sides to the ratio of the segments created on the opposite side by the bisector. Maybe there's an analogous theorem in three dimensions for dihedral angles?Let me think. In three dimensions, the concept might involve volumes or areas instead of just lengths. Since we're dealing with a dihedral angle, which is between two planes, the bisector plane should divide the tetrahedron in some proportional way.Perhaps I can use the concept of areas of the triangles (ACD) and (BCD). The areas are given as (t_{ACDDelta}) and (t_{BCDDelta}). Maybe the ratio of these areas affects where the bisector plane intersects edge (AB).I should consider the volumes of the smaller tetrahedrons formed by the bisector plane. If the bisector plane intersects edge (AB) at (E), then it divides the original tetrahedron into two smaller tetrahedrons: one with vertices (A), (C), (D), (E) and the other with vertices (B), (C), (D), (E).Let me denote the volumes of these smaller tetrahedrons as (V_1) and (V_2). If I can express these volumes in terms of the areas of triangles (ACD) and (BCD), and the lengths (AE) and (BE), then maybe I can find the desired ratio.The volume of a tetrahedron can be calculated using the formula:[V = frac{1}{3} times text{Base Area} times text{Height}]In this case, for tetrahedron (ACDE), the base is triangle (ACD) with area (t_{ACDDelta}), and the height would be the distance from point (E) to the plane containing triangle (ACD). Similarly, for tetrahedron (BCDE), the base is triangle (BCD) with area (t_{BCDDelta}), and the height is the distance from point (E) to the plane containing triangle (BCD).But wait, since the bisector plane is equidistant from both faces (ACD) and (BCD), the heights from (E) to both planes should be equal. Let me denote this common height as (h).Therefore, the volumes of the two smaller tetrahedrons can be written as:[V_1 = frac{1}{3} times t_{ACDDelta} times h][V_2 = frac{1}{3} times t_{BCDDelta} times h]Since the heights are the same, the ratio of the volumes (V_1) and (V_2) is simply the ratio of the areas of the bases:[frac{V_1}{V_2} = frac{t_{ACDDelta}}{t_{BCDDelta}}]But how does this relate to the ratio (frac{AE}{BE})?I think I need to express the volumes (V_1) and (V_2) in terms of (AE) and (BE). Let me consider the entire tetrahedron (ABCD). Its volume can be split into (V_1) and (V_2), so:[V_{ABCD} = V_1 + V_2]But I also know that the entire tetrahedron can be considered as two smaller tetrahedrons (ACDE) and (BCDE) combined. However, I need a way to relate the volumes (V_1) and (V_2) to the segments (AE) and (BE).Perhaps I can use the concept of similar triangles or some proportionality. Since the bisector plane intersects edge (AB) at (E), and given that the heights from (E) to both planes (ACD) and (BCD) are equal, there might be a proportional relationship between (AE), (BE), and the areas of the triangles.Let me denote the length (AE) as (x) and (BE) as (y). Therefore, the total length (AB = x + y).Now, considering the volumes (V_1) and (V_2), they are proportional to (x) and (y) respectively, but also scaled by the areas of the triangles (ACD) and (BCD). So, I can write:[V_1 = k times x times t_{ACDDelta}][V_2 = k times y times t_{BCDDelta}]where (k) is some constant of proportionality.Since the ratio of the volumes is equal to the ratio of the areas:[frac{V_1}{V_2} = frac{t_{ACDDelta}}{t_{BCDDelta}}]Substituting the expressions for (V_1) and (V_2):[frac{k times x times t_{ACDDelta}}{k times y times t_{BCDDelta}} = frac{t_{ACDDelta}}{t_{BCDDelta}}]The (k) terms cancel out, and we are left with:[frac{x}{y} = frac{t_{ACDDelta}}{t_{BCDDelta}}]Which means:[frac{AE}{BE} = frac{t_{ACDDelta}}{t_{BCDDelta}}]So, that's the result I was supposed to show.Wait, let me double-check if I missed anything. I considered the volumes as proportional to the segments (AE) and (BE) multiplied by the areas of the respective triangles. Since the heights from (E) to both planes are equal, the constant (k) should indeed be the same for both volumes. Therefore, the ratio simplifies correctly to the ratio of the areas.I think this makes sense. The bisector plane divides the edge (AB) in the ratio of the areas of the two adjacent faces. It's similar to the angle bisector theorem in two dimensions, but extended to three dimensions with areas instead of lengths.So, in summary, by considering the volumes of the smaller tetrahedrons formed by the bisector plane and recognizing that the heights from point (E) to both planes are equal, we can establish the proportional relationship between (AE), (BE), and the areas of triangles (ACD) and (BCD). This leads us directly to the desired ratio.**Final Answer**boxed{frac{AE}{BE} = frac{t_{ACDDelta}}{t_{BCDDelta}}}

answer：Alright, so I've got this problem with two parts, both involving propositions p and q. Let me try to understand what each part is asking and figure out how to solve them step by step.Starting with part (1): If a=1, and both propositions p and q are true, find the range of real number x.Okay, so first, let's parse what proposition p is. It says: For real number x, x² - 4a x + 3a² < 0, where a > 0. Since a is given as 1, I can substitute that in. So, p becomes x² - 4*1*x + 3*(1)² < 0, which simplifies to x² - 4x + 3 < 0.Now, I need to solve this quadratic inequality. To do that, I'll factor the quadratic. Let's see, x² - 4x + 3 can be factored into (x - 1)(x - 3). So, the inequality becomes (x - 1)(x - 3) < 0.To solve this inequality, I remember that the product of two terms is negative when one is positive and the other is negative. So, I need to find the intervals where this product is negative. The roots of the equation (x - 1)(x - 3) = 0 are x = 1 and x = 3. These roots divide the real number line into three intervals: (-∞, 1), (1, 3), and (3, ∞).Now, I'll test each interval to see where the product is negative.1. For x < 1, say x = 0: (0 - 1)(0 - 3) = (-1)(-3) = 3, which is positive.2. For 1 < x < 3, say x = 2: (2 - 1)(2 - 3) = (1)(-1) = -1, which is negative.3. For x > 3, say x = 4: (4 - 1)(4 - 3) = (3)(1) = 3, which is positive.So, the inequality (x - 1)(x - 3) < 0 holds true for x in (1, 3). Therefore, when a=1, proposition p is true for x in (1, 3).Now, moving on to proposition q. It says: For real number x, the system of inequalities:- |x - 1| ≤ 2- (x + 3)/(x - 2) ≥ 0I need to solve this system to find the range of x where q is true.First, let's solve |x - 1| ≤ 2. The absolute value inequality |x - 1| ≤ 2 means that x is within 2 units of 1. So, this translates to:-2 ≤ x - 1 ≤ 2Adding 1 to all parts:-2 + 1 ≤ x ≤ 2 + 1-1 ≤ x ≤ 3So, the first inequality gives us x in [-1, 3].Next, let's solve (x + 3)/(x - 2) ≥ 0. To solve this inequality, I need to find where the expression is positive or zero. The expression is a rational function, and it's undefined at x = 2 because the denominator becomes zero.To find the critical points, set the numerator and denominator equal to zero:- Numerator: x + 3 = 0 ⇒ x = -3- Denominator: x - 2 = 0 ⇒ x = 2These points divide the real number line into intervals: (-∞, -3), (-3, 2), and (2, ∞). I'll test each interval to determine the sign of the expression in that interval.1. For x < -3, say x = -4: (-4 + 3)/(-4 - 2) = (-1)/(-6) = 1/6, which is positive.2. For -3 < x < 2, say x = 0: (0 + 3)/(0 - 2) = 3/(-2) = -1.5, which is negative.3. For x > 2, say x = 3: (3 + 3)/(3 - 2) = 6/1 = 6, which is positive.Now, we also need to consider the points where the expression is zero or undefined:- At x = -3: The numerator is zero, so the expression is zero, which satisfies the inequality (since it's ≥ 0).- At x = 2: The expression is undefined, so x = 2 is excluded.Therefore, the solution to (x + 3)/(x - 2) ≥ 0 is x ≤ -3 or x > 2.Now, combining this with the first inequality |x - 1| ≤ 2, which gives x in [-1, 3], we need to find the intersection of x ≤ -3 or x > 2 with x in [-1, 3].Looking at x ≤ -3: The intersection with [-1, 3] is empty because -3 is less than -1, and there's no overlap.Looking at x > 2: The intersection with [-1, 3] is (2, 3].So, the solution to proposition q is x in (2, 3].Now, for part (1), we need both p and q to be true. So, we need the intersection of the ranges where p is true and q is true.From p: x in (1, 3)From q: x in (2, 3]The intersection is (2, 3).Therefore, the range of real number x is (2, 3).Moving on to part (2): If q is a sufficient but not necessary condition for p, find the range of real number a.Hmm, okay. So, q being a sufficient condition for p means that if q is true, then p is necessarily true. But it's not necessary, meaning p can be true without q being true.In logical terms, q ⇒ p, but p does not necessarily imply q.So, to translate this into our problem, the solution set of q must be a subset of the solution set of p. But since it's not necessary, the solution set of p must be larger than the solution set of q.From part (1), we know that when a=1, the solution set of p is (1, 3), and the solution set of q is (2, 3]. So, in that case, q is a subset of p, but since a=1 is just a specific case, we need to generalize this for any a > 0.First, let's recall the solution sets for p and q in terms of a.For proposition p: x² - 4a x + 3a² < 0.As before, factoring this quadratic: (x - a)(x - 3a) < 0.Since a > 0, the roots are x = a and x = 3a, with a < 3a.So, the solution set for p is (a, 3a).For proposition q, as before, the solution set is (2, 3].Wait, actually, hold on. In part (1), we had a=1, so the solution set for q was (2, 3]. But in general, is the solution set for q dependent on a? Wait, no, proposition q is defined as:For real number x, the system:- |x - 1| ≤ 2- (x + 3)/(x - 2) ≥ 0So, actually, proposition q is independent of a. It's always the same system, so its solution set is always (2, 3], regardless of a.Therefore, for any a > 0, the solution set for q is (2, 3].Now, for q to be a sufficient condition for p, the solution set of q must be a subset of the solution set of p. That is, (2, 3] must be a subset of (a, 3a).Additionally, since it's not necessary, (a, 3a) must be larger than (2, 3], meaning there are elements in (a, 3a) that are not in (2, 3].So, to ensure that (2, 3] is a subset of (a, 3a), we need:1. a ≤ 2: Because the lower bound of p's solution set is a, and to include 2, a must be less than or equal to 2.2. 3a > 3: Because the upper bound of p's solution set is 3a, and to include 3, 3a must be greater than 3. So, 3a > 3 ⇒ a > 1.Therefore, combining these two conditions:1. a ≤ 22. a > 1So, the range of a is (1, 2].But wait, let me double-check. If a=2, then the solution set for p is (2, 6). And q is (2, 3]. So, (2, 3] is indeed a subset of (2, 6). So, a=2 is acceptable.If a=1, then p is (1, 3), and q is (2, 3]. So, (2, 3] is a subset of (1, 3). But in part (2), we are told that q is a sufficient but not necessary condition for p. So, when a=1, p's solution set is (1, 3), and q is (2, 3]. So, q is a subset of p, but p is not a subset of q. So, q is sufficient but not necessary for p. So, a=1 is acceptable?Wait, but earlier, we concluded that a must be greater than 1. So, a=1 is not included in our range. Hmm, maybe I made a mistake.Wait, let's think again. If a=1, then p is (1, 3), and q is (2, 3]. So, q is a subset of p, which means q ⇒ p. So, q is a sufficient condition for p. But is it not necessary? That is, is there an x such that p is true but q is not? Yes, for example, x=1.5 is in p but not in q. So, q is sufficient but not necessary for p when a=1.But in our earlier reasoning, we said that to have (2, 3] ⊆ (a, 3a), we need a ≤ 2 and 3a > 3 ⇒ a > 1. But if a=1, then 3a = 3, so the upper bound is exactly 3. So, p is (1, 3), and q is (2, 3]. So, (2, 3] is still a subset of (1, 3). So, a=1 should be acceptable.Wait, but if a=1, then 3a=3, so p is (1, 3). But q is (2, 3]. So, 3 is included in q but not in p. Wait, no, p is x² -4a x +3a² < 0, which is strict inequality. So, at x=3, p is 9 - 12 + 3 = 0, which is not less than 0. So, x=3 is not included in p. But in q, x=3 is included because |3 -1| = 2 ≤ 2, and (3 +3)/(3 -2)=6/1=6 ≥0. So, x=3 is in q but not in p.Therefore, if a=1, q is (2, 3], and p is (1, 3). So, q is a subset of p except for the point x=3. But since p does not include x=3, and q does, is q still a subset of p? No, because x=3 is in q but not in p. Therefore, q is not entirely contained in p when a=1.Wait, that's a problem. So, if a=1, then q is (2, 3], but p is (1, 3). So, x=3 is in q but not in p. Therefore, q is not entirely contained in p. So, q is not a subset of p when a=1. Therefore, q is not a sufficient condition for p when a=1.Wait, but in part (1), when a=1, we found that the intersection of p and q is (2,3). So, in that case, both p and q being true gives x in (2,3). But for q to be a sufficient condition for p, q must imply p. That is, whenever q is true, p is also true. But when a=1, q includes x=3, which is not in p. So, q does not imply p when a=1.Therefore, a=1 is not acceptable for part (2). So, our earlier conclusion that a must be greater than 1 is correct.So, to ensure that (2, 3] is entirely contained within (a, 3a), we need:1. a ≤ 2: So that the lower bound of p is less than or equal to 2, ensuring that all x in (2, 3] are greater than a.2. 3a > 3: So that the upper bound of p is greater than 3, ensuring that all x in (2, 3] are less than 3a.Therefore, combining these two conditions:1. a ≤ 22. a > 1So, the range of a is (1, 2].Let me verify this with a=1.5. Then, p is (1.5, 4.5). q is (2, 3]. So, (2, 3] is a subset of (1.5, 4.5). Therefore, q implies p. And since p includes numbers less than 2 and greater than 3, which are not in q, q is not necessary for p. So, this works.Similarly, for a=2, p is (2, 6). q is (2, 3]. So, (2, 3] is a subset of (2, 6). And p includes numbers greater than 3, so q is not necessary for p. So, a=2 is acceptable.If a were greater than 2, say a=3, then p would be (3, 9). But q is (2, 3]. So, (2, 3] is not a subset of (3, 9), because 2 < 3. Therefore, q would not imply p, which violates the condition. So, a cannot be greater than 2.If a were less than or equal to 1, say a=0.5, then p would be (0.5, 1.5). But q is (2, 3], which is not a subset of (0.5, 1.5). So, q does not imply p, which again violates the condition.Therefore, the range of a must be (1, 2].So, to summarize:1. For part (1), when a=1, the range of x is (2, 3).2. For part (2), the range of a is (1, 2].

answer：Okay, so I have this geometry problem here with two planes, α and β, and two lines, m and n. I need to figure out which of the propositions A, B, C, or D is correct. Let me go through each option one by one and see if I can understand them properly.Starting with **Option A**: It says that if line m is parallel to plane α and line n is contained in plane α, then m is parallel to n. Hmm, let me visualize this. If m is parallel to α, that means m doesn't intersect α and lies entirely outside of it, right? And n is inside α. So, could m and n be parallel? Well, they could be, but they might also be skew lines. Skew lines are lines that don't intersect and aren't parallel, but they exist in different planes. Since m is parallel to α and n is in α, m and n could be skew. So, it's not necessarily true that m is parallel to n. Therefore, Option A might not always be correct.Moving on to **Option B**: This one states that if α is parallel to β, m is parallel to α, and n is parallel to β, then m is parallel to n. Okay, so α and β are parallel planes. If m is parallel to α, it means m doesn't intersect α and lies outside of it. Similarly, n is parallel to β, so it doesn't intersect β and lies outside of it. But since α and β are parallel, could m and n be parallel? They might be, but they could also be skew or even intersecting if they're in different orientations. For example, imagine two parallel planes, and two lines that are each parallel to their respective planes but not necessarily parallel to each other. They could cross each other somewhere else. So, I don't think we can guarantee that m is parallel to n just based on this information. So, Option B might not always be correct either.Now, **Option C**: It says that if m is perpendicular to β and n is parallel to β, then m is perpendicular to n. Let me think about this. If m is perpendicular to plane β, that means m is perpendicular to every line in β. And if n is parallel to β, then n is essentially lying in a plane that's parallel to β. So, if n is parallel to β, it's like n is in a plane that's a copy of β. Since m is perpendicular to β, it should also be perpendicular to any line parallel to β, right? Because the direction of m is orthogonal to the entire plane β, so any line parallel to β would also be orthogonal to m. That makes sense. So, I think Option C is correct.Finally, **Option D**: This one is a bit more complex. It says that if α is perpendicular to β, α intersects β at line m, and n is perpendicular to m, then n is perpendicular to α. Let me break this down. α and β are perpendicular, so their dihedral angle is 90 degrees. They intersect along line m, which is their line of intersection. Now, if n is perpendicular to m, does that mean n is perpendicular to α? Hmm, not necessarily. Because n being perpendicular to m only tells us about its relationship to the line of intersection, but not necessarily to the entire plane α. For n to be perpendicular to α, it would need to be perpendicular to every line in α, not just m. So, n could be in a different orientation relative to α. It might intersect α at some angle other than 90 degrees. Therefore, I don't think we can conclude that n is perpendicular to α just based on being perpendicular to m. So, Option D might not always be correct.Putting it all together, Option A isn't always true because m and n could be skew. Option B isn't necessarily true because m and n could be skew or intersecting. Option C seems correct because if m is perpendicular to β and n is parallel to β, then m must be perpendicular to n. Option D isn't necessarily true because n being perpendicular to m doesn't guarantee it's perpendicular to the entire plane α.So, after considering each option, I believe the correct answer is **Option C**.

answer：Okay, so I have this problem about a parabola with its vertex at the origin, and the focus F is on the positive y-axis. There are two parts: part I is to find the standard equation of the parabola, and part II is about finding the equation of a specific line m that intersects the parabola under certain conditions. Let me try to tackle part I first.Starting with part I. I remember that the standard form of a parabola with vertex at the origin and focus on the y-axis is ( x^2 = 4py ), where p is the distance from the vertex to the focus. So, if I can find p, I can write the equation.The problem says that a line passes through the focus F and intersects the parabola at points A and B. The length of segment AB is 8, and the distance from the midpoint of AB to the x-axis is 3. Hmm, okay, so let me visualize this. The parabola opens upwards since the focus is on the positive y-axis. A line goes through F and cuts the parabola at A and B. The midpoint of AB is 3 units above the x-axis, so its y-coordinate is 3.Let me denote the equation of the parabola as ( x^2 = 4py ). The focus F is at (0, p). Let the line passing through F have some slope, say m, so its equation is ( y = m x + p ). This line intersects the parabola at points A and B. To find these points, I can substitute ( y = m x + p ) into the parabola equation:( x^2 = 4p(m x + p) )( x^2 = 4p m x + 4p^2 )( x^2 - 4p m x - 4p^2 = 0 )This is a quadratic in x. Let me denote the roots as ( x_1 ) and ( x_2 ), which correspond to the x-coordinates of points A and B. From quadratic theory, I know that:( x_1 + x_2 = 4p m )( x_1 x_2 = -4p^2 )Now, the length of AB is 8. The distance between points A and B can be found using the distance formula. Let me denote A as ( (x_1, y_1) ) and B as ( (x_2, y_2) ). Then,( AB = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = 8 )But since both A and B lie on the line ( y = m x + p ), we can express ( y_2 - y_1 = m(x_2 - x_1) ). Therefore,( AB = sqrt{(x_2 - x_1)^2 + (m(x_2 - x_1))^2} = |x_2 - x_1| sqrt{1 + m^2} = 8 )So, ( |x_2 - x_1| sqrt{1 + m^2} = 8 ). Let me square both sides to eliminate the square root:( (x_2 - x_1)^2 (1 + m^2) = 64 )I know that ( (x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2 ). From earlier, ( x_1 + x_2 = 4p m ) and ( x_1 x_2 = -4p^2 ). Plugging these in:( (4p m)^2 - 4(-4p^2) = 16p^2 m^2 + 16p^2 = 16p^2(m^2 + 1) )So, ( (x_2 - x_1)^2 = 16p^2(m^2 + 1) ). Plugging back into the equation:( 16p^2(m^2 + 1)(1 + m^2) = 64 )( 16p^2(m^2 + 1)^2 = 64 )Divide both sides by 16:( p^2(m^2 + 1)^2 = 4 )Take square roots:( p(m^2 + 1) = 2 )So, ( p = frac{2}{m^2 + 1} )Okay, that's one equation involving p and m. Now, the other condition is that the midpoint of AB is 3 units from the x-axis, which means its y-coordinate is 3. The midpoint M has coordinates ( left( frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2} right) ). We already know ( x_1 + x_2 = 4p m ), so the x-coordinate of M is ( 2p m ). The y-coordinate is ( frac{y_1 + y_2}{2} ). Since both points are on the line ( y = m x + p ), we can write:( y_1 = m x_1 + p )( y_2 = m x_2 + p )So, ( y_1 + y_2 = m(x_1 + x_2) + 2p = m(4p m) + 2p = 4p m^2 + 2p )Therefore, ( frac{y_1 + y_2}{2} = 2p m^2 + p )We are told this is equal to 3:( 2p m^2 + p = 3 )Factor out p:( p(2m^2 + 1) = 3 )So, ( p = frac{3}{2m^2 + 1} )Now, from earlier, we have two expressions for p:1. ( p = frac{2}{m^2 + 1} )2. ( p = frac{3}{2m^2 + 1} )Set them equal:( frac{2}{m^2 + 1} = frac{3}{2m^2 + 1} )Cross-multiplying:( 2(2m^2 + 1) = 3(m^2 + 1) )Expand both sides:( 4m^2 + 2 = 3m^2 + 3 )Subtract 3m² and 2 from both sides:( m^2 = 1 )So, ( m = pm 1 )Now, plug back m² = 1 into one of the expressions for p:Using ( p = frac{2}{m^2 + 1} ):( p = frac{2}{1 + 1} = 1 )So, p = 1. Therefore, the standard equation of the parabola is ( x^2 = 4py = 4*1*y = 4y ). So, ( x^2 = 4y ).Wait, let me double-check. If p = 1, then the focus is at (0,1). The line through F with slope m = 1 or -1. Let me verify the length AB and the midpoint.Take m = 1. The line is y = x + 1. Substitute into parabola:( x^2 = 4(x + 1) )( x^2 - 4x - 4 = 0 )Solutions: ( x = [4 ± sqrt(16 + 16)]/2 = [4 ± sqrt(32)]/2 = [4 ± 4*sqrt(2)]/2 = 2 ± 2*sqrt(2) )So, points A and B have x-coordinates 2 + 2√2 and 2 - 2√2. Their y-coordinates are (2 + 2√2) + 1 = 3 + 2√2 and (2 - 2√2) + 1 = 3 - 2√2.Distance AB: sqrt[( (2 + 2√2 - (2 - 2√2))² + ( (3 + 2√2 - (3 - 2√2))² )]Simplify:Difference in x: 4√2, difference in y: 4√2So, distance AB: sqrt[(4√2)^2 + (4√2)^2] = sqrt[32 + 32] = sqrt[64] = 8. That's correct.Midpoint y-coordinate: (3 + 2√2 + 3 - 2√2)/2 = 6/2 = 3. Correct.Similarly, for m = -1, the line is y = -x + 1. Substitute into parabola:( x^2 = 4(-x + 1) )( x^2 + 4x - 4 = 0 )Solutions: ( x = [-4 ± sqrt(16 + 16)]/2 = [-4 ± sqrt(32)]/2 = [-4 ± 4√2]/2 = -2 ± 2√2 )Points A and B: (-2 + 2√2, y) and (-2 - 2√2, y). Compute y:For x = -2 + 2√2: y = -(-2 + 2√2) + 1 = 2 - 2√2 + 1 = 3 - 2√2For x = -2 - 2√2: y = -(-2 - 2√2) + 1 = 2 + 2√2 + 1 = 3 + 2√2Midpoint y-coordinate: (3 - 2√2 + 3 + 2√2)/2 = 6/2 = 3. Correct.Distance AB: same as before, 8. So, both slopes m = 1 and m = -1 satisfy the conditions. But since p is determined uniquely as 1, the equation is ( x^2 = 4y ).So, part I is solved, and the standard equation is ( x^2 = 4y ).Moving on to part II. This seems more complex. Let me read it again."Suppose a line m has a y-intercept of 6 and intersects the parabola at points P and Q. Connect QF and extend it to intersect the parabola's directrix at point R. When line PR is tangent to the parabola, find the equation of line m."Alright, so line m has equation y = kx + 6, where k is the slope we need to find. It intersects the parabola ( x^2 = 4y ) at points P and Q. Then, we connect Q to F (which is (0,1)) and extend that line to meet the directrix at R. The directrix of the parabola ( x^2 = 4y ) is y = -1. So, R is on y = -1.Then, when line PR is tangent to the parabola, we need to find the equation of line m, which is y = kx + 6.Let me break this down step by step.First, find points P and Q where line m intersects the parabola. Substitute y = kx + 6 into ( x^2 = 4y ):( x^2 = 4(kx + 6) )( x^2 - 4k x - 24 = 0 )Let me denote the roots as x3 and x4, corresponding to points P and Q. So,x3 + x4 = 4kx3 x4 = -24So, points P and Q are (x3, kx3 + 6) and (x4, kx4 + 6).Next, connect Q to F (0,1) and extend to meet the directrix y = -1 at R. So, the line QF is a line from Q(x4, kx4 + 6) to F(0,1). Let me find the equation of line QF.First, compute the slope of QF:Slope m1 = (1 - (k x4 + 6)) / (0 - x4) = (-k x4 -5)/(-x4) = (k x4 + 5)/x4So, the equation of line QF is:( y - 1 = m1 (x - 0) )( y = m1 x + 1 )( y = left( frac{k x4 + 5}{x4} right) x + 1 )We need to find where this line intersects the directrix y = -1. So, set y = -1:( -1 = left( frac{k x4 + 5}{x4} right) x + 1 )( -2 = left( frac{k x4 + 5}{x4} right) x )( x = frac{-2 x4}{k x4 + 5} )So, point R has coordinates:( left( frac{-2 x4}{k x4 + 5}, -1 right) )Now, we need to find the equation of line PR and find when it is tangent to the parabola.Point P is (x3, kx3 + 6), and point R is ( left( frac{-2 x4}{k x4 + 5}, -1 right) ). Let me denote R as (r_x, -1).So, line PR connects (x3, kx3 + 6) and (r_x, -1). The slope of PR is:( m2 = frac{-1 - (k x3 + 6)}{r_x - x3} = frac{-k x3 -7}{r_x - x3} )The equation of line PR is:( y - (k x3 + 6) = m2 (x - x3) )We need this line to be tangent to the parabola ( x^2 = 4y ). For a line to be tangent to a parabola, it must satisfy the condition that when substituted into the parabola equation, the discriminant is zero.Alternatively, the equation of the tangent to the parabola at a point (x0, y0) is ( x x0 = 2(y + y0) ). But since PR is tangent, it must touch the parabola at exactly one point. Let me use the condition that the system of equations (line PR and parabola) has exactly one solution.So, let me write the equation of line PR in terms of y:( y = m2 x + (k x3 + 6 - m2 x3) )Let me denote the equation as ( y = m2 x + c ), where c = ( k x3 + 6 - m2 x3 ).Substitute into parabola:( x^2 = 4(m2 x + c) )( x^2 - 4 m2 x - 4 c = 0 )For tangency, discriminant D = 0:( ( -4 m2 )^2 - 4 * 1 * (-4 c ) = 0 )( 16 m2^2 + 16 c = 0 )Divide by 16:( m2^2 + c = 0 )So, ( c = -m2^2 )Therefore, ( k x3 + 6 - m2 x3 = -m2^2 )Rearrange:( k x3 + 6 = m2 x3 - m2^2 )( k x3 + 6 = m2 (x3 - m2) )But m2 is the slope of PR, which we have as ( frac{-k x3 -7}{r_x - x3} ). Also, r_x is ( frac{-2 x4}{k x4 + 5} ).This is getting quite involved. Maybe there's a better approach.Alternatively, since PR is tangent to the parabola, let me denote the point of tangency as T(t, t²/4). The equation of the tangent at T is ( x t = 2(y + t²/4) ), which simplifies to ( x t = 2y + t²/2 ), or ( 2y = x t - t²/2 ), so ( y = (x t)/2 - t²/4 ).Since PR is this tangent line, it must pass through points P and R. So, substituting P(x3, kx3 + 6) into the tangent equation:( kx3 + 6 = (x3 t)/2 - t²/4 )Similarly, substituting R(r_x, -1):( -1 = (r_x t)/2 - t²/4 )So, we have two equations:1. ( kx3 + 6 = (x3 t)/2 - t²/4 )2. ( -1 = (r_x t)/2 - t²/4 )Also, from earlier, we have expressions for x3 + x4 = 4k and x3 x4 = -24.Additionally, r_x is expressed in terms of x4: ( r_x = frac{-2 x4}{k x4 + 5} )This is a system of equations with variables x3, x4, t, k. It's quite complicated, but maybe we can find relations between them.Let me try to express t from equation 2:From equation 2:( -1 = frac{r_x t}{2} - frac{t^2}{4} )Multiply both sides by 4:( -4 = 2 r_x t - t^2 )Rearrange:( t^2 - 2 r_x t - 4 = 0 )Similarly, from equation 1:( kx3 + 6 = frac{x3 t}{2} - frac{t^2}{4} )Multiply both sides by 4:( 4k x3 + 24 = 2 x3 t - t^2 )Rearrange:( t^2 - 2 x3 t + 4k x3 + 24 = 0 )So now, we have two quadratic equations in t:1. ( t^2 - 2 r_x t - 4 = 0 )2. ( t^2 - 2 x3 t + 4k x3 + 24 = 0 )Since both equal zero, set them equal to each other:( t^2 - 2 r_x t - 4 = t^2 - 2 x3 t + 4k x3 + 24 )Cancel t²:( -2 r_x t - 4 = -2 x3 t + 4k x3 + 24 )Bring all terms to one side:( -2 r_x t - 4 + 2 x3 t - 4k x3 - 24 = 0 )Factor terms:( t(2 x3 - 2 r_x) - 4k x3 - 28 = 0 )Divide both sides by 2:( t(x3 - r_x) - 2k x3 - 14 = 0 )So,( t = frac{2k x3 + 14}{x3 - r_x} )But r_x is ( frac{-2 x4}{k x4 + 5} ), so:( x3 - r_x = x3 + frac{2 x4}{k x4 + 5} )This is getting too convoluted. Maybe I need a different approach.Alternatively, since PR is tangent, the condition is that the system of PR and the parabola has exactly one solution. So, let me write the equation of PR and set discriminant to zero.Equation of PR: passes through P(x3, kx3 + 6) and R(r_x, -1). Let me find its equation.Slope m2 = (-1 - (k x3 + 6)) / (r_x - x3) = (-k x3 -7)/(r_x - x3)Equation: y - (k x3 + 6) = m2 (x - x3)Express y:y = m2 x - m2 x3 + k x3 + 6So, y = m2 x + (k x3 - m2 x3 + 6)Now, substitute into parabola:x² = 4y = 4(m2 x + k x3 - m2 x3 + 6)So,x² - 4 m2 x - 4(k x3 - m2 x3 + 6) = 0For tangency, discriminant D = 0:( -4 m2 )² - 4 * 1 * (-4(k x3 - m2 x3 + 6)) = 016 m2² + 16(k x3 - m2 x3 + 6) = 0Divide by 16:m2² + k x3 - m2 x3 + 6 = 0So,m2² - m2 x3 + k x3 + 6 = 0But m2 = (-k x3 -7)/(r_x - x3)Let me denote m2 = (-k x3 -7)/(r_x - x3) = (-k x3 -7)/(- (x3 - r_x)) = (k x3 +7)/(x3 - r_x)So, m2 = (k x3 +7)/(x3 - r_x)Let me denote D = x3 - r_x, so m2 = (k x3 +7)/DPlugging into the equation:[(k x3 +7)/D]^2 - [(k x3 +7)/D] x3 + k x3 + 6 = 0Multiply through by D² to eliminate denominators:(k x3 +7)^2 - (k x3 +7) x3 D + (k x3 +6) D² = 0This is getting too messy. Maybe I need to find another relation.Recall that x3 and x4 are roots of x² -4k x -24=0, so x3 + x4 =4k, x3 x4 = -24.Also, r_x = (-2 x4)/(k x4 +5)Let me express r_x in terms of x4, and since x4 = 4k - x3 (from x3 + x4 =4k), maybe substitute that.So, x4 =4k -x3Thus,r_x = (-2 (4k -x3))/(k (4k -x3) +5) = (-8k + 2x3)/(4k² -k x3 +5)So, r_x = (2x3 -8k)/(4k² -k x3 +5)Thus, x3 - r_x = x3 - (2x3 -8k)/(4k² -k x3 +5)Let me compute x3 - r_x:= [x3 (4k² -k x3 +5) - (2x3 -8k)] / (4k² -k x3 +5)= [4k² x3 -k x3² +5x3 -2x3 +8k] / denominator= [4k² x3 -k x3² +3x3 +8k] / denominatorSimilarly, m2 = (k x3 +7)/(x3 - r_x) = (k x3 +7) * [denominator]/[4k² x3 -k x3² +3x3 +8k]This is getting too complicated. Maybe instead of trying to express everything in terms of x3, I can use symmetric sums.Given that x3 + x4 =4k and x3 x4 = -24, perhaps I can express everything in terms of k.Let me denote S = x3 + x4 =4k, P =x3 x4 = -24.Also, note that r_x = (-2 x4)/(k x4 +5). Let me express r_x in terms of x4.But x4 = S -x3 =4k -x3.So,r_x = (-2 (4k -x3))/(k (4k -x3) +5) = (-8k +2x3)/(4k² -k x3 +5)Let me denote numerator as N = -8k +2x3, denominator as D =4k² -k x3 +5.So, r_x = N/D.Now, let me express m2:m2 = (k x3 +7)/(x3 - r_x) = (k x3 +7)/(x3 - N/D) = (k x3 +7)/[(D x3 - N)/D] = D(k x3 +7)/(D x3 - N)Compute D x3 - N:= (4k² -k x3 +5) x3 - (-8k +2x3)=4k² x3 -k x3² +5x3 +8k -2x3=4k² x3 -k x3² +3x3 +8kSo, m2 = D(k x3 +7)/(4k² x3 -k x3² +3x3 +8k)But D =4k² -k x3 +5, so:m2 = (4k² -k x3 +5)(k x3 +7)/(4k² x3 -k x3² +3x3 +8k)This is still complicated, but maybe I can find a relation.Recall that from the tangency condition, we had:m2² - m2 x3 +k x3 +6 =0Let me plug m2 into this equation.But this seems too involved. Maybe instead, I can use parametric forms.Alternatively, perhaps using the property that PR is tangent, so the point of tangency T lies on both PR and the parabola, and the tangent condition holds.Alternatively, maybe using parametric equations for the parabola. For parabola ( x^2 =4y ), parametric equations are x=2pt, y=pt², but since p=1, it's x=2t, y=t².So, any point on the parabola can be written as (2t, t²). The tangent at this point is x t = y + t², or y = t x - t².So, the tangent line at (2t, t²) is y = t x - t².Since PR is this tangent line, it must pass through both P and R.Point P is (x3, kx3 +6). So, substituting into tangent equation:k x3 +6 = t x3 - t²Similarly, point R is (r_x, -1). Substituting into tangent equation:-1 = t r_x - t²So, we have two equations:1. k x3 +6 = t x3 - t²2. -1 = t r_x - t²From equation 1:t² - t x3 +k x3 +6=0From equation 2:t² - t r_x -1=0Set them equal since both equal t²:t x3 -k x3 -6 = t r_x +1So,t(x3 - r_x) = k x3 +7Thus,t = (k x3 +7)/(x3 - r_x)But from earlier, r_x = (-2 x4)/(k x4 +5). And x4 =4k -x3.So,r_x = (-2 (4k -x3))/(k (4k -x3) +5) = (-8k +2x3)/(4k² -k x3 +5)Thus,x3 - r_x =x3 - (-8k +2x3)/(4k² -k x3 +5) = [x3 (4k² -k x3 +5) +8k -2x3]/(4k² -k x3 +5)= [4k² x3 -k x3² +5x3 +8k -2x3]/(4k² -k x3 +5)= [4k² x3 -k x3² +3x3 +8k]/(4k² -k x3 +5)So,t = (k x3 +7) * (4k² -k x3 +5)/(4k² x3 -k x3² +3x3 +8k)This is still complicated, but maybe I can find a relation between t and x3.Alternatively, let me consider that t is a parameter, and perhaps express x3 in terms of t.From equation 1:k x3 +6 = t x3 - t²Rearrange:x3 (t -k) = t² +6So,x3 = (t² +6)/(t -k)Similarly, from equation 2:-1 = t r_x - t²So,r_x = (t² -1)/tBut r_x is also equal to (-2 x4)/(k x4 +5). And x4 =4k -x3.So,r_x = (-2 (4k -x3))/(k (4k -x3) +5) = (-8k +2x3)/(4k² -k x3 +5)But we have r_x = (t² -1)/t, so:(-8k +2x3)/(4k² -k x3 +5) = (t² -1)/tCross-multiplying:(-8k +2x3) t = (t² -1)(4k² -k x3 +5)But x3 = (t² +6)/(t -k). Let me substitute that:(-8k +2*(t² +6)/(t -k)) t = (t² -1)(4k² -k*(t² +6)/(t -k) +5)This is very involved, but let me try to simplify step by step.First, compute left side:= [ -8k + (2t² +12)/(t -k) ] * t= [ (-8k(t -k) +2t² +12) / (t -k) ] * t= [ (-8k t +8k² +2t² +12) / (t -k) ] * t= t*(-8k t +8k² +2t² +12)/(t -k)Similarly, compute right side:= (t² -1)(4k² -k*(t² +6)/(t -k) +5)= (t² -1)[4k² +5 -k*(t² +6)/(t -k)]Let me compute the term inside the brackets:=4k² +5 - [k(t² +6)]/(t -k)So,= [ (4k² +5)(t -k) -k(t² +6) ] / (t -k)= [4k² t -4k³ +5t -5k -k t² -6k ] / (t -k)= [ -k t² +4k² t -k t² +5t -4k³ -11k ] / (t -k)Wait, let me expand (4k² +5)(t -k):=4k² t -4k³ +5t -5kThen subtract k(t² +6):=4k² t -4k³ +5t -5k -k t² -6k= -k t² +4k² t +5t -4k³ -11kSo, right side becomes:(t² -1) * [ (-k t² +4k² t +5t -4k³ -11k ) / (t -k) ]Thus, overall equation:t*(-8k t +8k² +2t² +12)/(t -k) = (t² -1)*(-k t² +4k² t +5t -4k³ -11k ) / (t -k)Multiply both sides by (t -k):t*(-8k t +8k² +2t² +12) = (t² -1)*(-k t² +4k² t +5t -4k³ -11k )Let me expand both sides.Left side:= t*(-8k t +8k² +2t² +12)= -8k t² +8k² t +2t³ +12tRight side:= (t² -1)*(-k t² +4k² t +5t -4k³ -11k )Multiply term by term:= t²*(-k t² +4k² t +5t -4k³ -11k ) -1*(-k t² +4k² t +5t -4k³ -11k )= -k t^4 +4k² t^3 +5t^3 -4k³ t² -11k t² +k t² -4k² t -5t +4k³ +11kSimplify:- k t^4 +4k² t^3 +5t^3 -4k³ t² -11k t² +k t² -4k² t -5t +4k³ +11kCombine like terms:- k t^4 + (4k² +5) t^3 + (-4k³ -11k +k) t² + (-4k² -5) t + (4k³ +11k)= -k t^4 + (4k² +5) t^3 + (-4k³ -10k) t² + (-4k² -5) t +4k³ +11kNow, set left side equal to right side:-8k t² +8k² t +2t³ +12t = -k t^4 + (4k² +5) t^3 + (-4k³ -10k) t² + (-4k² -5) t +4k³ +11kBring all terms to left side:0 = -k t^4 + (4k² +5) t^3 + (-4k³ -10k) t² + (-4k² -5) t +4k³ +11k +8k t² -8k² t -2t³ -12tSimplify term by term:- t^4 term: -k t^4t^3 terms: (4k² +5 -2) t^3 = (4k² +3) t^3t² terms: (-4k³ -10k +8k) t² = (-4k³ -2k) t²t terms: (-4k² -5 -8k² -12) t = (-12k² -17) tConstants:4k³ +11kSo, overall equation:- k t^4 + (4k² +3) t^3 + (-4k³ -2k) t² + (-12k² -17) t +4k³ +11k =0This is a quartic equation in t, which is very complicated. Maybe I can factor out k:= -k t^4 +4k² t^3 +3 t^3 -4k³ t² -2k t² -12k² t -17 t +4k³ +11kHmm, not obvious. Maybe factor by grouping.Alternatively, perhaps assume specific values for k? Since in part I, p=1, and the parabola is x²=4y, maybe the line m has a simple slope, like k=1/2 or something.Alternatively, maybe the slope k is such that the quartic equation factors nicely.Alternatively, perhaps there's a smarter approach.Wait, maybe instead of going through all this, I can use the condition that PR is tangent, so the point of tangency T must satisfy that the line PR is the tangent at T, and T lies on PR.So, T is (2t, t²), and PR passes through P(x3, kx3 +6) and R(r_x, -1). So, the slope of PR must be equal to the slope of the tangent at T, which is t.So, slope of PR is ( -1 - (k x3 +6) ) / (r_x -x3 ) = (-k x3 -7)/(r_x -x3 ) = tSo,(-k x3 -7)/(r_x -x3 ) = tBut r_x = (-2 x4)/(k x4 +5), and x4=4k -x3So,r_x = (-2(4k -x3))/(k(4k -x3)+5) = (-8k +2x3)/(4k² -k x3 +5)Thus,(-k x3 -7)/( (-8k +2x3)/(4k² -k x3 +5) -x3 ) = tSimplify denominator:= [ (-8k +2x3) -x3(4k² -k x3 +5) ] / (4k² -k x3 +5 )= [ -8k +2x3 -4k² x3 +k x3² -5x3 ] / (4k² -k x3 +5 )= [ -8k -3x3 -4k² x3 +k x3² ] / (4k² -k x3 +5 )So,(-k x3 -7) * (4k² -k x3 +5 ) / [ -8k -3x3 -4k² x3 +k x3² ] = tBut from earlier, we have:From equation 1: k x3 +6 = t x3 - t² => t² - t x3 +k x3 +6=0From equation 2: -1 = t r_x - t² => t² - t r_x -1=0So, we have two equations:1. t² - t x3 +k x3 +6=02. t² - t r_x -1=0Subtract equation 2 from equation 1:(t² - t x3 +k x3 +6) - (t² - t r_x -1)=0Simplify:- t x3 +k x3 +6 + t r_x +1=0= t(r_x -x3 ) +k x3 +7=0But from earlier, t = (-k x3 -7)/(r_x -x3 )So,t(r_x -x3 ) = -k x3 -7Thus,(-k x3 -7) +k x3 +7=0 => 0=0Which is an identity, so no new information.This suggests that the system is dependent, and we need another condition.Alternatively, perhaps using the fact that x3 and x4 are roots of x² -4k x -24=0, so x3 +x4=4k, x3 x4=-24.Also, from equation 1: t² - t x3 +k x3 +6=0From equation 2: t² - t r_x -1=0Subtracting equation 2 from equation 1:- t x3 +k x3 +6 + t r_x +1=0= t(r_x -x3 ) +k x3 +7=0But t = (-k x3 -7)/(r_x -x3 )So,[ (-k x3 -7)/(r_x -x3 ) ]*(r_x -x3 ) +k x3 +7=0Simplifies to:(-k x3 -7) +k x3 +7=0 => 0=0Again, no new info.This suggests that the equations are consistent but not independent, so we need another relation.Perhaps using the fact that x3 x4 = -24, and x4=4k -x3.So,x3 (4k -x3 ) = -244k x3 -x3² = -24x3² -4k x3 -24=0Which is consistent with the quadratic we had earlier.So, maybe express x3 in terms of k:x3 = [4k ± sqrt(16k² +96)]/2 = 2k ± sqrt(4k² +24)But this might not help directly.Alternatively, let me consider that from equation 1:t² = t x3 -k x3 -6From equation 2:t² = t r_x +1Set equal:t x3 -k x3 -6 = t r_x +1So,t(x3 - r_x ) =k x3 +7But t = (-k x3 -7)/(r_x -x3 )So,[ (-k x3 -7)/(r_x -x3 ) ]*(x3 - r_x ) =k x3 +7Simplify:(-k x3 -7)*(-1) =k x3 +7Which is:k x3 +7 =k x3 +7Again, identity.This suggests that the system is dependent, and we need another approach.Perhaps, instead of trying to solve for t and x3, I can use the fact that the tangent line PR must satisfy certain properties.Alternatively, maybe consider specific values for k.Given that in part I, the slope m was ±1, maybe in part II, the slope k is ±1/2 or something similar.Let me test k=1/2.If k=1/2, then the line m is y=(1/2)x +6.Find intersection with parabola x²=4y:x²=4*( (1/2)x +6 )=2x +24x² -2x -24=0Solutions: x=(2 ±sqrt(4 +96))/2=(2±sqrt(100))/2=(2±10)/2=6 or -4So, points P(6, (1/2)*6 +6=9) and Q(-4, (1/2)*(-4)+6=4)So, P(6,9), Q(-4,4)Now, connect Q(-4,4) to F(0,1). The line QF has slope (1-4)/(0 -(-4))= (-3)/4= -3/4Equation of QF: y -1= (-3/4)(x -0)=> y= (-3/4)x +1Find intersection R with directrix y=-1:-1= (-3/4)x +1 => (-3/4)x= -2 => x= (-2)*(-4/3)=8/3So, R(8/3, -1)Now, find line PR connecting P(6,9) and R(8/3, -1)Slope m2= (-1 -9)/(8/3 -6)= (-10)/(8/3 -18/3)= (-10)/(-10/3)=3Equation of PR: y -9=3(x -6)=> y=3x -18 +9=3x -9Check if this is tangent to parabola x²=4y.Substitute y=3x -9 into x²=4y:x²=12x -36x² -12x +36=0Discriminant: 144 -144=0Yes, it's tangent. So, k=1/2 works.Similarly, test k=-1/2.Line m: y= (-1/2)x +6Intersect with parabola:x²=4*(-1/2 x +6)= -2x +24x² +2x -24=0Solutions: x=(-2 ±sqrt(4 +96))/2=(-2±10)/2=4 or -6Points P(4, (-1/2)*4 +6=4) and Q(-6, (-1/2)*(-6)+6=9)So, P(4,4), Q(-6,9)Connect Q(-6,9) to F(0,1). Slope=(1-9)/(0 -(-6))= (-8)/6= -4/3Equation of QF: y -1= (-4/3)(x -0)=> y= (-4/3)x +1Find R on directrix y=-1:-1= (-4/3)x +1 => (-4/3)x= -2 => x= (-2)*(-3/4)=3/2So, R(3/2, -1)Line PR connects P(4,4) and R(3/2, -1)Slope m2= (-1 -4)/(3/2 -4)= (-5)/(-5/2)=2Equation of PR: y -4=2(x -4)=> y=2x -8 +4=2x -4Check tangency:Substitute y=2x -4 into x²=4y:x²=8x -16x² -8x +16=0Discriminant:64 -64=0Yes, tangent. So, k=-1/2 also works.Therefore, the possible slopes k are ±1/2, so the equations of line m are y= (1/2)x +6 and y= (-1/2)x +6.Thus, the equations are y= ±(1/2)x +6.**Final Answer**(I) The standard equation of the parabola is boxed{x^2 = 4y}.(II) The equation of line ( m ) is boxed{y = pm frac{1}{2}x + 6}.