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Question: A circle of radius r is concentric with an ellipse\(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}\) = 1....

A circle of radius r is concentric with an ellipsex2a2+y2b2\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1. If common tangent is inclined to the major axis at an angle of q, then tan2q equals-

A

r2−b2a2−b2\frac{r^{2} - b^{2}}{a^{2} - b^{2}}

B

r2−b2a2−r2\frac{r^{2} - b^{2}}{a^{2} - r^{2}}

C

r2−b2r2−a2\frac{r^{2} - b^{2}}{r^{2} - a^{2}}

D

r2−a2b2−r2\frac{r^{2} - a^{2}}{b^{2} - r^{2}}

Answer

r2−b2a2−r2\frac{r^{2} - b^{2}}{a^{2} - r^{2}}

Explanation

Solution

equation of ellipse

x2a2+y2b2\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 ̃ Equation of circle x2 + y2 = r2

Equation of tangent to ellipse

̃ y = mx ± a2m2+b2\sqrt{a^{2}m^{2} + b^{2}}

Equation of tangent to circle

̃ y = mx ± r 1+m2\sqrt{1 + m^{2}}

for common tangent

± r 1+m2\sqrt{1 + m^{2}} = ± a2m2+b2\sqrt{a^{2}m^{2} + b^{2}}

r2 + r2m2 = a2 m2 + b2 ̃ a2m2 – r2m2 = r2 – b2

m2 = r2–b2a2–r2\frac{r^{2}–b^{2}}{a^{2}–r^{2}} ̃ tan2q = r2–b2a2–r2\frac{r^{2}–b^{2}}{a^{2}–r^{2}}