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Question

The locus of the image of the focus of the ellipse x225+y29=1 with respect to any of the tangents to the ellipse is

A
(x+4)2+y2=100
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B
(x+2)2+y2=50
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C
(x4)2+y2=100
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D
(x2)2+y2=50
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Solution

The correct options are
A (x+4)2+y2=100
C (x4)2+y2=100

Equation of the ellipse is
x225+y29=1
Foci (±4,0)
Let S(h,k) be the image.
Mid point of SS is M(h±42,k2)
SS cuts tangent line at point M which lies on the auxiliary circle of the ellipse , since foot of perpendicular from foci upon any tangent lies on auxilary circle.
Auxiliary circle of the ellipse is x2+y2=25
M lies on auxiliary circle.
So,
(h±42)2+k24=25
Required locus equation is (x±4)2+y2=100

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