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The tangents to the parabola y^2=4a x at...

The tangents to the parabola `y^2=4a x` at the vertex `V` and any point `P` meet at `Q` . If `S` is the focus, then prove that `S PdotS Q ,` and `S V` are in GP.

A

`A.P.`

B

`G.P.`

C

`H.P.`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let `y^(2)=4ax` be a parabola with vertex at A(0, 0) and `P(at^(2), 2at)` be any point on it. The equation of tangents at A and P are x=0 and `ty=x+at^(3)` respectively. These two intersect at Q(0, at). The focus S of dsthe parabola `y^(2)=4ax` has coordinates (a, 0).
`:." "SP=a+at^(2),SQ=ssqrt(1+t^(2))" and "SA=a` ltrbgt `rArr" "SQ^(2)=SPxxSArArrSP, SQ, SA" are in G.P."`
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OBJECTIVE RD SHARMA ENGLISH-PARABOLA-Chapter Test
  1. The tangents to the parabola y^2=4a x at the vertex V and any point P ...

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  2. If y=2x+k is a tangent to the curve x^(2)=4y, then k is equal to

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  3. The normal drawn at a point (a t1 2,-2a t1) of the parabola y^2=4a x m...

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  4. The mid-point of the chord 2x+y-4=0 of the parabola y^(2)=4x is

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  5. The two ends of latusrectum of a parabola are the points (3, 6) and (-...

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  6. Prove that the locus of the middle points of all chords of the parabol...

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  7. The focus of the parabola x^2-8x+2y+7=0 is

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  8. The point of contact of the line x-2y-1=0 with the parabola y^(2)=2(x-...

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  9. Find the number of distinct normals that can be drawn from (-2,1) to t...

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  10. At what point on the parabola y^2=4x the normal makes equal angle with...

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  11. Three normals to the parabola y^2= x are drawn through a point (C, O) ...

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  12. The normal chord of a parabola y^2= 4ax at the point P(x1, x1) subten...

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  13. AB, AC are tangents to a parabola y^2=4ax; p1, p2, p3 are the lengths...

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  14. The circles on the focal radii of a parabola as diameter touch: A) th...

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  15. If the normals from any point to the parabola y^2=4x cut the line x=2 ...

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  16. about to only mathematics

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  17. The equation of the tangent to the parabola y^(2)=8x which is perpendi...

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  18. the tangent drawn at any point P to the parabola y^2= 4ax meets the di...

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  19. about to only mathematics

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  20. The parabola y^(2)=4ax passes through the point (2,-6). Find the lengt...

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  21. A variable circle passes through the fixed point (2, 0) and touches y-...

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