Q: The equation of common tangents to the ellipse x^{2} + 2y^{2} = 1 and the circle x^{2} + y^{2} = 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 x^{2} + y^{2} = 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.