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The straight lines joining any point `P` on the parabola `y^2=4a x` to the vertex and perpendicular from the focus to the tangent at `P` intersect at `Rdot` Then the equation of the locus of `R` is

A

`x^(2)+2y^(2)-ax=0`

B

`2x^(2)+y^(2)-ax=0`

C

`2x^(2)+2y^(2)-ay=0`

D

`2x^(2)+y^(2)-ay=0`

Text Solution

Verified by Experts

The correct Answer is:
B
(2) Tangent at point P is `ty=x+at^(2)`.
The line perpendicular to (1) passes through (a,0). Therefore,
y-0=-t(x-1)
`ortx+y=ta`
`ory=t(a-x)` (1)
The equation of OP is
`y-(2)/(t)x=0ory=(2)/(t)x` (3)
From (2) and (3), eliminating
t,we get
`y^(2)=2x(a-x)`
`or2x^(2)+y^(2)ax=0`
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CENGAGE-PARABOLA-Exercise (Single)
  1. The tangent to y^(2)=ax make angles theta(1)andtheta(2) with the x-axi...

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  2. A tangent is drawn to the parabola y^2=4 x at the point P whose abscis...

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  3. The straight lines joining any point P on the parabola y^2=4a x to the...

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  4. Through the vertex O of the parabola y^2=4a x , two chords O Pa n dO Q...

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  5. A B is a double ordinate of the parabola y^2=4a xdot Tangents drawn to...

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  6. If the locus of the middle of point of contact of tangent drawn to the...

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  7. If the bisector of angle A P B , where P Aa n dP B are the tangents to...

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  8. From a point A(t) on the parabola y^(2)=4ax, a focal chord and a tange...

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  9. The point of intersection of the tangents of the parabola y^(2)=4x dra...

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  10. The angle between the tangents to the parabola y^2=4a x at the points ...

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  11. y=x+2 is any tangent to the parabola y^2=8xdot The point P on this tan...

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  12. If y=m1x+c and y=m2x+c are two tangents to the parabola y^2+4a(x+a)=0 ...

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  13. The angle between the tangents to the curve y=x^2-5x+6 at the point (2...

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  14. Two mutually perpendicular tangents of the parabola y^(2)=4ax meet the...

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  15. Radius of the circle that passes through the origin and touches the ...

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  16. The mirror image of the parabola y^2=4x in the tangent to the parab...

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  17. Consider the parabola y^2=4xdot Let A-=(4,-4) and B-=(9,6) be two fixe...

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  18. A line of slope lambda(0 lt lambda lt 1) touches the parabola y+3x^2=0...

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  19. The tangent at any point P onthe parabola y^2=4a x intersects the y-ax...

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  20. If P(t^2,2t),t in [0,2] , is an arbitrary point on the parabola y^2=4x...

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