The tangent drawn from $(\alpha,\ \beta)$ to an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ touches the circle $x^{2} + y^{2} = c^{2}$, then the locus of $(\alpha,\ \beta)$ is

Question

Dr. Abubacker · Accepted Answer

(a)
The chord of contact of $(\alpha,\ \beta)$ is $\frac{x\alpha}{a^{2}} + \frac{y\beta}{b^{2}} = 1$. It touches the circle $x^{2} + y^{2} = c^{2}$
$	herefore\frac{1}{\sqrt{\frac{\alpha^{2}}{a^{4}} + \frac{\beta^{2}}{b^{4}}}} = c$
$\Rightarrow \frac{\alpha^{2}}{a^{4}} + \frac{\beta^{2}}{b^{4}} = \frac{1}{c^{2}}$
Thus, the locus of $(\alpha,\ \beta)$ is
$\frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}} = \frac{1}{c^{2}}$

Q. The tangent drawn from $(\alpha,\ \beta)$ to an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ touches the circle $x^{2} + y^{2} = c^{2}$, then the locus of $(\alpha,\ \beta)$ is

Explanation

Reviewed by Dr. Abubacker

Want more practice?