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Question

If tangents are drawn to the ellipse x2+2y2=2 at all points on the ellipse other than its four vertices, then the mid points of the tangents intercepted between the coordinates axes lie on the curve :

A
12x2+14y2=1
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B
14x2+12y2=1
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C
x22+y24=1
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D
x24+y22=1
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Solution

The correct option is A 12x2+14y2=1
Equation of tangent to ellipse x2+2y2=2 at (2cosθ,sinθ) is xcosθ2+ysinθ1=1
Therefore, it intersects co-ordinate axes at (2cosθ,0) and (0,1sinθ)
Mid point (h,k) is
h=12cosθcosθ=12h
k=12sinθsinθ=12k

sin2θ+cos2θ=1
12h2+14k2=1
Locus is 12x2+14y2=1

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