Question

A circle of radius 2 is concentric with ellipse $\frac{x^2}{7}+\frac{y^2}{3}=1$ then inclination of common tangent with x-axis -

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is D $\frac{\pi }{6}$

Given,

a circle $c,x^2+y^2=4$

equation of tangent of ellipse.

$y=mx+\sqrt{7m^2+3}$

equation of tangent of ellipse

$y=mx\pm 2\sqrt{1+m^2}$

Equating equation

(i) and (ii)

$mx+\sqrt{7m^2+3}=mx\pm 2\sqrt{1+m^2}$

Squaring both sides

$(7m^2+3)=4(1+m^2)$

$7m^2+3\ne 4+4m^2$

$3m^2=1$

$m^2=\frac{1}{3}$

$m=\pm \frac{1}{\sqrt{3}}$

$m=\tan \theta =\frac{1}{\sqrt{3}}$

$\therefore \theta =\frac{\pi }{6}$