Question

The slopes of the common tangents of the ellipse $\frac{x^2}{9}+\frac{y^2}{2}=1$ and the circle $x^2+y^2=3$ are

• A. $\pm \frac{1}{\sqrt{3}}$
• B. $\pm \frac{1}{\sqrt{2}}$
• C. $\pm \frac{1}{\sqrt{6}}$
• D. None of these

Answer 1

The correct option is C $\pm \frac{1}{\sqrt{6}}$

Let $m$ be the slope of the common tangent, then
$\pm \sqrt{3}\sqrt{1+m^2}=\pm \sqrt{9m^2+2}$
$\Rightarrow 3(1+m^2)=9m^2+2$
$\Rightarrow 3+3m^2=9m^2+2$
$\Rightarrow 6m^2=1$
$\Rightarrow m^2=\frac{1}{6}$
$\Rightarrow m=\pm \frac{1}{\sqrt{6}}$