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PN is an ordinate of the parabola y2 = 4ax. A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex in a point T such that AT = kNP, then the value of k is (where A is the vertex)
- a)3/2
- b)2/3
- c)1
- d)none
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
PN is an ordinate of the parabola y2= 4ax. A straight line is drawn pa...
The equation of parabola be y2=4ax
let the point P be (at2,2at)
PN is ordinate ⇒N(at2,0)
Equation of straight line bisecting NP is
y=at
substituting y in equation of parabola
a2t2 = 4ax
⇒x = 4at2
So the coordinates of Q are (4at2 ,at)
Equation of NQ is y−0 = (at−0)/(at2/4 - at2)(x−at2)
y= −4/3t(x−at2)
Put x=0
y = −4/3t(0−at^2)
⇒y=4at/3
⇒AT = 4at/3
NP = 2at
AT/NP = (4at/3)/2at
= ⅔
AT = 2/3NP
let the point P be (at2,2at)
PN is ordinate ⇒N(at2,0)
Equation of straight line bisecting NP is
y=at
substituting y in equation of parabola
a2t2 = 4ax
⇒x = 4at2
So the coordinates of Q are (4at2 ,at)
Equation of NQ is y−0 = (at−0)/(at2/4 - at2)(x−at2)
y= −4/3t(x−at2)
Put x=0
y = −4/3t(0−at^2)
⇒y=4at/3
⇒AT = 4at/3
NP = 2at
AT/NP = (4at/3)/2at
= ⅔
AT = 2/3NP
Community Answer
PN is an ordinate of the parabola y2= 4ax. A straight line is drawn pa...
Given:
- Parabola equation: y^2 = 4ax
- PN is an ordinate of the parabola
- A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q
- NQ meets the tangent at the vertex in a point T such that AT = kNP
- A is the vertex
To find: The value of k
Solution:
1. Drawing the diagram
- Let's draw a diagram to better visualize the problem.
- We know that PN is an ordinate of the parabola, which means it is perpendicular to the x-axis.
- Also, a line is drawn parallel to the axis of the parabola to bisect NP. Let's call this line MN.
- MN intersects the parabola at point Q.
- NQ meets the tangent at the vertex, which means it passes through the vertex. Let's call this point of intersection T.
- We need to find the value of k, given that AT = kNP.
- Here's a rough sketch of the diagram:
T
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V (vertex)
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Q |
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N |
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P * * * * * * * |
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2. Finding the coordinates of P and N
- Since PN is perpendicular to the x-axis, we know that its x-coordinate is the same as that of Q. Let's call this x-coordinate a.
- We also know that PN is bisected by line MN. Let's call the midpoint of PN as M.
- Since M is the midpoint of PN, its y-coordinate is the average of the y-coordinates of P and N. Let's call the y-coordinate of M as b.
- Therefore, the coordinates of P are (a, 2b) and the coordinates of N are (a, 0).
3. Finding the coordinates of Q
- We know that Q lies on the parabola y^2 = 4ax and also on the line MN.
- Since MN is parallel to the x-axis, we know that the y-coordinate of Q is b.
- To find the x-coordinate of Q, we can substitute y = b in the equation of the parabola:
b^2 = 4ax
a = b^2/4x
Putting this value of a in x = a, we get:
x = b^2/4a
- Therefore, the coordinates of Q are (b^2/4a, b).
4. Finding the equation of the tangent at the vertex
- The vertex of the parabola is at (0, 0).
- The equation of the parabola is y^2 = 4ax.
- To find the equation of the tangent at the vertex, we need to find the derivative of y with respect to x:
dy/dx = (1/2)(4ax)^(-1/2) * 4a
= 2a^(-1/2)
- At the vertex (0, 0), the derivative is infinite. Therefore, the equation of the tangent at the vertex is
- Parabola equation: y^2 = 4ax
- PN is an ordinate of the parabola
- A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q
- NQ meets the tangent at the vertex in a point T such that AT = kNP
- A is the vertex
To find: The value of k
Solution:
1. Drawing the diagram
- Let's draw a diagram to better visualize the problem.
- We know that PN is an ordinate of the parabola, which means it is perpendicular to the x-axis.
- Also, a line is drawn parallel to the axis of the parabola to bisect NP. Let's call this line MN.
- MN intersects the parabola at point Q.
- NQ meets the tangent at the vertex, which means it passes through the vertex. Let's call this point of intersection T.
- We need to find the value of k, given that AT = kNP.
- Here's a rough sketch of the diagram:
T
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V (vertex)
|
|
|
|
|
Q |
* |
|
N |
* |
|
P * * * * * * * |
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2. Finding the coordinates of P and N
- Since PN is perpendicular to the x-axis, we know that its x-coordinate is the same as that of Q. Let's call this x-coordinate a.
- We also know that PN is bisected by line MN. Let's call the midpoint of PN as M.
- Since M is the midpoint of PN, its y-coordinate is the average of the y-coordinates of P and N. Let's call the y-coordinate of M as b.
- Therefore, the coordinates of P are (a, 2b) and the coordinates of N are (a, 0).
3. Finding the coordinates of Q
- We know that Q lies on the parabola y^2 = 4ax and also on the line MN.
- Since MN is parallel to the x-axis, we know that the y-coordinate of Q is b.
- To find the x-coordinate of Q, we can substitute y = b in the equation of the parabola:
b^2 = 4ax
a = b^2/4x
Putting this value of a in x = a, we get:
x = b^2/4a
- Therefore, the coordinates of Q are (b^2/4a, b).
4. Finding the equation of the tangent at the vertex
- The vertex of the parabola is at (0, 0).
- The equation of the parabola is y^2 = 4ax.
- To find the equation of the tangent at the vertex, we need to find the derivative of y with respect to x:
dy/dx = (1/2)(4ax)^(-1/2) * 4a
= 2a^(-1/2)
- At the vertex (0, 0), the derivative is infinite. Therefore, the equation of the tangent at the vertex is
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PN is an ordinate of the parabola y2= 4ax. A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex in a point T such that AT = kNP, then the value of k is (where A is the vertex)a)3/2b)2/3c)1d)noneCorrect answer is option 'B'. Can you explain this answer?
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PN is an ordinate of the parabola y2= 4ax. A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex in a point T such that AT = kNP, then the value of k is (where A is the vertex)a)3/2b)2/3c)1d)noneCorrect answer is option 'B'. 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 PN is an ordinate of the parabola y2= 4ax. A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex in a point T such that AT = kNP, then the value of k is (where A is the vertex)a)3/2b)2/3c)1d)noneCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PN is an ordinate of the parabola y2= 4ax. A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex in a point T such that AT = kNP, then the value of k is (where A is the vertex)a)3/2b)2/3c)1d)noneCorrect answer is option 'B'. Can you explain this answer?.
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