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Question

If tangent to parabola y2=4ax intersects ellipse x2a2+y2b2=1 at A and B, then the locus of point of intersection of tangents at A and B, is

A
straight line
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B
parabola with length of latus rectum b4a3 unit
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C
ellipse
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D
parabola with length of latus rectum a4b3 unit
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Solution

The correct option is B parabola with length of latus rectum b4a3 unit
Let P(h,k) be the point of intersection of tangents at A and B.
So, equation of chord of contact AB is
hxa2+kyb2=1(i)

Also, equation of tangent to parabola y2=4ax is
y=mx+ammxy=am(ii)

If equation (i) and (ii) are identical, then
mh/a2=1k/b2=a/m1
m=hb2ka2 and m=akb2
hb2ka2=akb2

Hence, required locus is
y2=b4a3x
which is a parabola, with length of latus rectum b4a3 units.

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