Question

A circle of radius 'r' is concentric with the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Then inclination of common tangent with major axis is ___ (b<r<a)

• A. $ta{n}^{-1}\left(\frac{b}{a}\right)$
• B. $ta{n}^{-1}\left(\frac{rb}{a}\right)$
• C. $ta{n}^{-1}\left(\frac{r^2-b^2}{a^2-r^2}\right)$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is C $ta{n}^{-1}\left(\frac{r^2-b^2}{a^2-r^2}\right)$

The tangent of Ellipse is $y=mx+\sqrt{a^2{m}^2+b^2}$, this line touches $x^2+y^2=r^2$
Condition is $\left|\frac{\sqrt{a^2{m}^2+b^2}}{\sqrt{m^2+1}}\right|=r$
$a^2{m}^2+b^2=r^2{m}^2+r^2$
$m^2(a^2-r^2)=r^2-b^2\Rightarrow m^2-\frac{r^2-b^2}{a^2-r^2}$
$m=\sqrt{\frac{r^2-b^2}{a^2-r^2}}$
Inclination is $ta{n}^{-1}\left(\frac{r^2-b^2}{a^2-r^2}\right)$.