Question

If a circle of radius r is concentric with ellipse ,then the common tangent is inclined to major axis is

Answer 1

Consider the equation of ellipse having center at (0,0) and major axis at (a,0) and (-a,0) respectively.

$\frac{x^2}{a^2} +\frac{y^2}{b^2}=1$

and equation of circle having radius r and center at (0,0)  is

x²+ y²=r²

Tangent to ellipse is given by differentiating the equation of ellipse  once with respect to x

$\frac{2 x}{a^2} +\frac{2 yy'}{b^2}=1\\\\ y'=\frac{-b^2 x}{a^2y}$

$y'_{(a,0)}=$∞

Let angle between two tangents be α.

Tan α = ∞

α= 90°

If tangent is drawn from circle to major axis

Differentiating equation of circle

2 x + 2 y y'=0

y'=  $\frac{-x}{y}$

Since it passes through (a,0) and (-a,0)

y'=∞

tan β= ∞

β=90°