Question

Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an ellipse.

Answer 1

Let P(θ₁) and Q(θ₂) be any two points on the given ellipse such that θ₁ – θ₂ = k, where k is a constant. 

The equation of the tangent at point P(θ₁) is

Multiplying equation (i) by cos θ₂ and equation (ii) by cos θ₁ and subtracting, we get y/b (sin θ₁ cos θ₂ – sin₂ θ cos θ₁ ) = cos θ₂ – cos θ₁

_{\(\frac {x^2}{a^2} + \frac {y^2}{b^2} = sec^2\,\frac{k}{2},\) which is an ellipse.}