Question

Equation of the locus of the pole with respect to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of any tangent line to the auxiliary circle is the curve
$\frac{x^2}{a^4}+\frac{y^2}{b^4}={\lambda }^2$ where

• A. ${\lambda }^2=a^2$
• B. ${\lambda }^2=\frac{1}{a^2}$
• C. ${\lambda }^2=b^2$
• D. ${\lambda }^2=\frac{1}{b^2}$

Answer 1

The correct option is B

${\lambda }^2=\frac{1}{a^2}$

Equation of the auxiliary circle is $x^2+y^2=a^2$ (i)
Let (h, k) be the pole, then the equation of the polar of (h, k) with respect to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is
$\frac{hx}{a^2}+\frac{ky}{b^2}=1$ (ii)
Since (ii) is a tangent to circle (i),
$\frac{-1}{\sqrt{{\left(\frac{h}{a^2}\right)}^2+{\left(\frac{k}{b^2}\right)}^2}}=\pm a\Rightarrow \frac{h^2}{a^4}+\frac{k^2}{b^4}=\frac{1}{a^2}\therefore Locusof(h,k)is\frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{a^2}$