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Question

The locus of the foot of perpendicular drawn from the centre of the ellipse x2+3y2=6 on any tangent to it is:

A
(x2y2)2=6x2+2y2
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B
(x2y2)2=6x22y2
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C
(x2+y2)2=6x2+2y2
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D
(x2+y2)2=6x22y2
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Solution

The correct option is C (x2+y2)2=6x2+2y2
Let the foot of perpendicular drawn from the centre of ellipse on the tangent l is (h,k)


Slope of OP=kh
Slope of tangent, m=hk
As we know, Equation of tangent is
y=mx±a2m2+b2...(1)
put value of m in equation (1) and pass through (h,k)
Thus, k=hkh±6(hk)2+2
k2+h2=±6h2+2k2
(k2+h2)2=6h2+2k2
Put (x,y) at the place of (h,k)
Hence, (x2+y2)2=6x2+2y2

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