Question

The slopes of the common tangents of the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ and circle $x^2+y^2=3$ are

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

Answer 1

The correct option is B $\pm \sqrt{2}$

Let $m$ be the slope of the commom tangent, then
$\pm \sqrt{3}\sqrt{1+m^2}=\pm \sqrt{4m^2+1}$
$\Rightarrow 3+3m^2=4m^2+1$
$\Rightarrow m^2=2$
$\Rightarrow m=\pm \sqrt{2}$