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Parabolas y^(2)=4a(x-c(1))andx^(2)=4a(y-...

Parabolas `y^(2)=4a(x-c_(1))andx^(2)=4a(y-c_(2))`, where `c_(1)` and `c_(2)` are variable, are such that they touch each other. The locus of their point of contact is

A

straight line

B

Circle

C

Paraabola

D

hyperbola

Text Solution

Verified by Experts

The correct Answer is:
D
Let `P(x_(1), y_(1))` be the point of contact of the two parabolas. Tangents at P to the two parabolas are `yy_(1)=2a(x+x_(1))-4alamda_(1)rArr2ax-yy_(1)=2a(2lamda_(1)-x_(1))" ....(i)"`
`"and "x x_(1)=2a(y+y_(1))-4alamda_(2)rArrx x_(1)-2ay=2a(y_(1)-2lamda_(2))" ...(ii)"`
Clearly (i) and (ii) represent the same line.
`:." "(2a)/(x_(1))=y_(1)/(2a)rArrx_(1)y_(2)=4a^(2)`
Hence, the locus of `(x_(1), y_(1))` is `xy=4a^(2)`, which is a hyperbola.
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OBJECTIVE RD SHARMA ENGLISH-PARABOLA-Chapter Test
  1. Parabolas y^(2)=4a(x-c(1))andx^(2)=4a(y-c(2)), where c(1) and c(2) are...

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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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