# The equation of common tangents to the ellipse x^2 + 2y^2 = 1 and the circle x^2 + y^2 = 2/3 is

Q: The equation of common tangents to the ellipse x2 + 2y2 = 1 and the circle x2 + y2 = 2/3 is

(A) $\large y = \frac{\sqrt{7}}{\sqrt{2}} x + \sqrt{3}$

(B) y = 7x + √3

(C) y = √7x + √2

(D) none of these

Sol. The equation of any tangent to the circle x2 + y2 = 2/3 is $\large y = \pm \frac{\sqrt{2}}{\sqrt{3}} \sqrt{1 + m^2}$ since it touches the given ellipse then

$\large \frac{\sqrt{2}}{\sqrt{3}} \sqrt{1 + m^2} = \sqrt{m^2 – \frac{1}{2}}$

Squaring ,

$\large \frac{2}{3} + \frac{2}{3}m^2 = m^2 -\frac{1}{2}$

$\large m = \pm \frac{\sqrt{7}}{\sqrt{2}}$

$\large y = \frac{\sqrt{7}}{\sqrt{2}} x + \sqrt{3}$ is the common tangent.