Question

If $\alpha -\beta =$ constant, then the locus of the point of intersection of tangents at $P(a\cos \alpha ,b\sin \alpha )$ and $Q(a\cos \beta ,b\cos \beta )$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

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

Answer 1

Answer: (C)

An ellipse

Let $R(h,k)$ be the point of intersection of tangents at $P$ and $Q$. Then,
$h=\frac{a\cos \left(\frac{\alpha +\beta }{2}\right)}{\cos \left(\frac{\alpha -\beta }{2}\right)}$
and $k=\frac{b\cos \left(\frac{\alpha +\beta }{2}\right)}{\cos \left(\frac{\alpha -\beta }{2}\right)}$
$\Rightarrow \frac{h^2}{a^2}+\frac{k^2}{b^2}=\frac{1}{{\cos }^2\left(\frac{\alpha -\beta }{2}\right)}$
Here, the locus of $R(h,k)$ is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{{\cos }^2\left(\frac{\alpha -\beta }{2}\right)}$
Clearly it represents an ellipse.