Question

The locus of the point of intersection of tangents to an ellipse at two points, sum of whose eccentric angles is constant, is a/an

• A. parabola
• B. circle
• C. ellipse
• D. straight line

Answer 1

Answer: (D)

As in the above question, the point of intersection is
$(h,k)\equiv \left(\frac{a\cos \left(\frac{\alpha +\beta }{2}\right)}{\cos \left(\frac{\alpha -\beta }{2}\right)},\frac{b\sin \left(\frac{\alpha +\beta }{2}\right)}{\cos \left(\frac{\alpha -\beta }{2}\right)}\right)$
It is given that $\alpha +\beta =c=$ constant.
Therefore, $h=\frac{a\cos \frac{c}{2}}{\cos \left(\frac{\alpha -\beta }{2}\right)}$
and $k=\frac{b\sin \frac{c}{2}}{\cos \left(\frac{\alpha -\beta }{2}\right)}$
or $\frac{h}{k}=\frac{a}{b}\cot \left(\frac{c}{2}\right)$
or $k=\frac{b}{a}\tan \left(\frac{c}{2}\right)h$
Therefore, $(h,k)$ lies on the straight line.