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The locus of the point of intersection o...

The locus of the point of intersection of tangents of `x^2/a^2 + y^2/b^2 =1` at two points whose eccentric angles differ by `pi/2` is an ellipse whose semi-axes are: (A) `sqrt2 a,sqrt 2 b` (B) `a/sqrt2,b/sqrt2` (C) `a/2,b/2` (D) `2a,2b`

Text Solution

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Let the points on ellipse be `p(acos(theta),bsin(theta))` and
`Q(acos(pi/2+theta),bsin(pi/2+theta))=(-asintheta,bcostheta)`
Equation of tangent at point p:
`x/acostheta+y/bsintheta=1`
Equation of tangent at point Q:
`-x/asintheta+y/bcostheta=1`
On solving these equations we get, ...
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