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Q.
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
Conic Sections
Solution:
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.