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Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?
- a)The triangle PFQ is a right-angled triangle.
- b)The triangle QPQ' is a right-angled triangle.
- c)The distance between P and F is 5√2.
- d)F lies on the line joining Q and Q'.
Correct answer is option 'A,B,D'. Can you explain this answer?
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Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be ...
First, we need to find the equation of the tangent at Q. We can use the fact that the tangent at a point on a parabola is perpendicular to the line joining the point and the focus. The focus of the given parabola is (2,0), so the line joining Q and F is given by y = -4x + 8. The slope of the tangent at Q is the negative reciprocal of -4, which is 1/4. The tangent at Q passes through Q, so its equation is y - 8 = 1/4(x - (-2)), which simplifies to y = 1/4x + 9.
Similarly, we can find the equation of the tangent at Q. The focus of the given parabola is (-2,0), so the line joining Q' and F is given by y = 4x + 8. The slope of the tangent at Q' is the negative reciprocal of 4, which is -1/4. The tangent at Q' passes through Q', so its equation is y - 4 = -1/4(x - (-2)), which simplifies to y = -1/4x + 5.
Now we can find the coordinates of Q and Q by solving the system of equations consisting of the equation of E and the equations of the tangents at Q and Q'. Substituting y = 1/4x + 9 into y2 = 8x, we get (1/4x + 9)2 = 8x, which simplifies to x = 4. Substituting x = 4 into y = 1/4x + 9, we get y = 10, so Q is (4,10). Similarly, substituting y = -1/4x + 5 into y2 = 8x, we get (-1/4x + 5)2 = 8x, which simplifies to x = 4. Substituting x = 4 into y = -1/4x + 5, we get y = 4, so Q' is (4,4).
Now we can use the distance formula to find the lengths of the sides of the triangles PFQ and PQ'Q. The focus F is (2,0), so PF = sqrt((2 - (-2))2 + (0 - 4)2) = sqrt(20) = 2sqrt(5). Using the distance formula, we find that QF = sqrt((4 - 2)2 + (10 - 0)2) = 2sqrt(26), and Q'F = sqrt((4 - (-2))2 + (4 - 0)2) = 2sqrt(20) = 4sqrt(5). Therefore, neither triangle is a right-angled triangle.
Finally, we can use the distance formula to find the distance between P and F. PF is already known to be 2sqrt(5), so we just need to verify that PF = 5. Squaring both sides of the equation PF = 2sqrt(5), we get PF2 = 20. Squaring both sides of the equation 5 = sqrt(25), we get 52 = 25. Therefore, PF2 = 20 = 52, which implies that PF = 5.
Similarly, we can find the equation of the tangent at Q. The focus of the given parabola is (-2,0), so the line joining Q' and F is given by y = 4x + 8. The slope of the tangent at Q' is the negative reciprocal of 4, which is -1/4. The tangent at Q' passes through Q', so its equation is y - 4 = -1/4(x - (-2)), which simplifies to y = -1/4x + 5.
Now we can find the coordinates of Q and Q by solving the system of equations consisting of the equation of E and the equations of the tangents at Q and Q'. Substituting y = 1/4x + 9 into y2 = 8x, we get (1/4x + 9)2 = 8x, which simplifies to x = 4. Substituting x = 4 into y = 1/4x + 9, we get y = 10, so Q is (4,10). Similarly, substituting y = -1/4x + 5 into y2 = 8x, we get (-1/4x + 5)2 = 8x, which simplifies to x = 4. Substituting x = 4 into y = -1/4x + 5, we get y = 4, so Q' is (4,4).
Now we can use the distance formula to find the lengths of the sides of the triangles PFQ and PQ'Q. The focus F is (2,0), so PF = sqrt((2 - (-2))2 + (0 - 4)2) = sqrt(20) = 2sqrt(5). Using the distance formula, we find that QF = sqrt((4 - 2)2 + (10 - 0)2) = 2sqrt(26), and Q'F = sqrt((4 - (-2))2 + (4 - 0)2) = 2sqrt(20) = 4sqrt(5). Therefore, neither triangle is a right-angled triangle.
Finally, we can use the distance formula to find the distance between P and F. PF is already known to be 2sqrt(5), so we just need to verify that PF = 5. Squaring both sides of the equation PF = 2sqrt(5), we get PF2 = 20. Squaring both sides of the equation 5 = sqrt(25), we get 52 = 25. Therefore, PF2 = 20 = 52, which implies that PF = 5.
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Community Answer
Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be ...
E = y2 - 8x = 0
a = 2
(1) (slope of PF) (slope of FQ) = -1
⇒ ∠PFQ = π/2
(2) (slope of PQ')(slope of PQ) = -1
⇒ ∠QPQ'= π/2
(3) PF = 445
(4) slope of QT = slope of FQ
⇒ Q, F, Q' are collinear
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Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer?
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Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer?.
Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer?.
Solutions for Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer?, a detailed solution for Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer? has been provided alongside types of Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let E denote the parabola y2 = 8x. Let P = (-2, 4) and let Q and Q be two distinct points on E such that the lines PQ and PQ are tangents to E. Let F be the focus of E. Then which of the following statements is/are TRUE?a)The triangle PFQ is a right-angled triangle.b)The triangle QPQ is a right-angled triangle.c)The distance between P and F is 5√2.d)F lies on the line joining Q and Q.Correct answer is option 'A,B,D'. Can you explain this answer? tests, examples and also practice JEE tests.
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