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A variable chord PQ of the parabola y = 4x2 substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q is
- a)4y + 1= 16x2
- b)y + 4 = 0
- c)4y + 4 = 4x2
- d)4y + 1 = 0
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
A variable chord PQ of the parabola y = 4x2 substends a right angle at...
Slope of OP x slope of OQ = -1
⇒ 4t1.4t2 = -1
Eq of tangent at
Eq of tangent at
Let (x1 , y1) is the point of intersection
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A variable chord PQ of the parabola y = 4x2 substends a right angle at...
To find the locus of points of intersection of the tangents at points P and Q on the parabola, we need to first determine the coordinates of P and Q.
Given that the variable chord PQ of the parabola y = 4x^2 subtends a right angle at the vertex, we can use the property that the product of the slopes of two perpendicular lines is -1.
Let the coordinates of P be (x1, 4x1^2) and the coordinates of Q be (x2, 4x2^2).
The slope of the line joining P and Q is (4x2^2 - 4x1^2) / (x2 - x1).
Since the chord PQ subtends a right angle at the vertex, the product of the slopes of the lines joining the vertex to P and Q is -1.
The slope of the line joining the vertex (0, 0) to P is (4x1^2 - 0) / (x1 - 0) = 4x1.
The slope of the line joining the vertex (0, 0) to Q is (4x2^2 - 0) / (x2 - 0) = 4x2.
Therefore, we have the equation:
4x1 * 4x2 = -1.
Simplifying this equation, we get:
16x1x2 = -1.
Now, we need to find the equation of the tangent at point P on the parabola.
The equation of the tangent at point P is given by y - 4x1^2 = 8x1(x - x1).
Simplifying this equation, we get:
y - 4x1^2 = 8x1x - 8x1^2.
y = 8x1x - 4x1^2.
Similarly, the equation of the tangent at point Q is:
y = 8x2x - 4x2^2.
To find the locus of points of intersection of the tangents at P and Q, we need to find the equation of the curve that satisfies both of these equations.
By equating the two equations, we get:
8x1x - 4x1^2 = 8x2x - 4x2^2.
Simplifying this equation, we get:
4x1^2 - 4x2^2 = 8x2x - 8x1x.
4(x1^2 - x2^2) = 8x2x - 8x1x.
Dividing both sides by 4, we get:
x1^2 - x2^2 = 2x2x - 2x1x.
x1^2 - x2^2 = 2x(x2 - x1).
Substituting the value of x2 - x1 from the equation 16x1x2 = -1, we get:
x1^2 - x2^2 = 2x(-1/16).
x1^2 - x2^2 = -x/8.
Multiplying both sides by -1, we get:
Given that the variable chord PQ of the parabola y = 4x^2 subtends a right angle at the vertex, we can use the property that the product of the slopes of two perpendicular lines is -1.
Let the coordinates of P be (x1, 4x1^2) and the coordinates of Q be (x2, 4x2^2).
The slope of the line joining P and Q is (4x2^2 - 4x1^2) / (x2 - x1).
Since the chord PQ subtends a right angle at the vertex, the product of the slopes of the lines joining the vertex to P and Q is -1.
The slope of the line joining the vertex (0, 0) to P is (4x1^2 - 0) / (x1 - 0) = 4x1.
The slope of the line joining the vertex (0, 0) to Q is (4x2^2 - 0) / (x2 - 0) = 4x2.
Therefore, we have the equation:
4x1 * 4x2 = -1.
Simplifying this equation, we get:
16x1x2 = -1.
Now, we need to find the equation of the tangent at point P on the parabola.
The equation of the tangent at point P is given by y - 4x1^2 = 8x1(x - x1).
Simplifying this equation, we get:
y - 4x1^2 = 8x1x - 8x1^2.
y = 8x1x - 4x1^2.
Similarly, the equation of the tangent at point Q is:
y = 8x2x - 4x2^2.
To find the locus of points of intersection of the tangents at P and Q, we need to find the equation of the curve that satisfies both of these equations.
By equating the two equations, we get:
8x1x - 4x1^2 = 8x2x - 4x2^2.
Simplifying this equation, we get:
4x1^2 - 4x2^2 = 8x2x - 8x1x.
4(x1^2 - x2^2) = 8x2x - 8x1x.
Dividing both sides by 4, we get:
x1^2 - x2^2 = 2x2x - 2x1x.
x1^2 - x2^2 = 2x(x2 - x1).
Substituting the value of x2 - x1 from the equation 16x1x2 = -1, we get:
x1^2 - x2^2 = 2x(-1/16).
x1^2 - x2^2 = -x/8.
Multiplying both sides by -1, we get:
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A variable chord PQ of the parabola y = 4x2 substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q isa)4y + 1= 16x2b)y + 4 = 0c)4y + 4 = 4x2d)4y + 1 = 0Correct answer is option 'D'. 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 A variable chord PQ of the parabola y = 4x2 substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q isa)4y + 1= 16x2b)y + 4 = 0c)4y + 4 = 4x2d)4y + 1 = 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A variable chord PQ of the parabola y = 4x2 substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q isa)4y + 1= 16x2b)y + 4 = 0c)4y + 4 = 4x2d)4y + 1 = 0Correct answer is option 'D'. Can you explain this answer?.
A variable chord PQ of the parabola y = 4x2 substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q isa)4y + 1= 16x2b)y + 4 = 0c)4y + 4 = 4x2d)4y + 1 = 0Correct answer is option 'D'. 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 A variable chord PQ of the parabola y = 4x2 substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q isa)4y + 1= 16x2b)y + 4 = 0c)4y + 4 = 4x2d)4y + 1 = 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A variable chord PQ of the parabola y = 4x2 substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q isa)4y + 1= 16x2b)y + 4 = 0c)4y + 4 = 4x2d)4y + 1 = 0Correct answer is option 'D'. Can you explain this answer?.
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