Question

Equation of the common tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and the circle $x^2+y^2=12$ is

• A. $2y=\sqrt{3}x+\sqrt{21}$
• B. $\sqrt{3}y=2x+\sqrt{21}$
• C. $\sqrt{3}y=2x-\sqrt{21}$
• D. $y$

Answer 1

The correct option is A $2y=\sqrt{3}x+\sqrt{21}$

Equation of tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ is $y=mx\pm \sqrt{16m^2+9}$
$\Rightarrow mx-y\pm \sqrt{16m^2+9}=0$
If it is also Tangent to the circle $x^2+y^2=12$ then r = d holds good,
$\therefore \sqrt{12}=\frac{\sqrt{16M^2+9}}{\sqrt{M^2+1}}$
$\Rightarrow 12(m^2+1)=16m^2+9$
$\Rightarrow m^2=\frac{3}{4}\Rightarrow m=\pm \frac{\sqrt{3}}{2}$
$\therefore Tangentis2.y=\sqrt{3}x\pm \sqrt{21}$