Question

Find the equation of common tangent to circle $x^2+y^2=5$ and ellipse $x^2+9y^2=9$.

Answer 1

The equation of any tangent to the circle $x^2+y^2=5$ is $y=mx+\sqrt{5(1+m^2)}$......(1) and that to the ellipse $x^2=9y^2=9$ is$y=mx+\sqrt{9m^2+1}$.......(2).

Now the tangents (1) and (2) will be the common one, if

$\sqrt{5(m^2+1)}=\sqrt{9m^2+1}$

or, $5m^2+5=9m^2+1$

or, $4m^2=4$

or, $m^2=1$

or, $m=\pm 1$

So, two common tangents are there and they are $y=x+\sqrt{10}$ and $y=-x+\sqrt{10}$.