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Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ is
- a)y2 = 2x + 8
- b)y2 = x + 8
- c)y2 = 2x – 8
- d)y2 = x – 8
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn...
Given parabola is y2 = 4x ...(1)
Let 
Slope of OP = 
and slope of OQ = 
Since OP 
OQ, 
OQ, Let R (h, k) be the middle point of PQ, then
and k = t1 + t2 ...(4)
From (4), 
[ From(2) and (3)]
Hence locus of R (h, k) is y2 = 2x – 8.
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Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn...
Let's call the midpoint of PQ as M, and let's say that the coordinates of M are (h, k).
Since O is the vertex of the parabola, the x-coordinate of O is 0. Therefore, the x-coordinate of P is also 0 (since OP is drawn at a right angle to OQ).
Since PQ is a chord of the parabola, it satisfies the equation y^2 = 4x. Plugging in the x-coordinate of P (which is 0), we get y^2 = 4(0) = 0. Therefore, the y-coordinate of P is 0.
Since M is the midpoint of PQ, the coordinates of M are the average of the coordinates of P and Q. Therefore, the y-coordinate of M is (0 + k)/2 = k/2.
Since M lies on the locus of the midpoint of PQ, we know that the y-coordinate of M squared is equal to 2 times its x-coordinate. Therefore, we have (k/2)^2 = 2h.
Simplifying this equation, we get k^2/4 = 2h. Multiplying both sides by 4, we get k^2 = 8h. Rewrite this equation in the form y^2 = 8x, we get y^2 = 8x.
Therefore, the locus of the midpoint of PQ is y^2 = 8x.
The correct answer is not listed among the options given.
Since O is the vertex of the parabola, the x-coordinate of O is 0. Therefore, the x-coordinate of P is also 0 (since OP is drawn at a right angle to OQ).
Since PQ is a chord of the parabola, it satisfies the equation y^2 = 4x. Plugging in the x-coordinate of P (which is 0), we get y^2 = 4(0) = 0. Therefore, the y-coordinate of P is 0.
Since M is the midpoint of PQ, the coordinates of M are the average of the coordinates of P and Q. Therefore, the y-coordinate of M is (0 + k)/2 = k/2.
Since M lies on the locus of the midpoint of PQ, we know that the y-coordinate of M squared is equal to 2 times its x-coordinate. Therefore, we have (k/2)^2 = 2h.
Simplifying this equation, we get k^2/4 = 2h. Multiplying both sides by 4, we get k^2 = 8h. Rewrite this equation in the form y^2 = 8x, we get y^2 = 8x.
Therefore, the locus of the midpoint of PQ is y^2 = 8x.
The correct answer is not listed among the options given.
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Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer?
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Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer?.
Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer?.
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Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer?, a detailed solution for Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of Through the vertex O of a parabola y2 = 4x, chords OP and OQ are drawn at right angles to one another. The locus of the middle point of PQ isa)y2 = 2x + 8b)y2 = x + 8c)y2 = 2x – 8d)y2 = x – 8Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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