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

A circle of radius ‘r’ is concentric with x2a2+y2b2\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}=1. The slope of a common tangent to them is

A

r2−b2\sqrt{r^{2} - b^{2}}

B

r2−a2\sqrt{r^{2} - a^{2}}

C

r2+b2\sqrt{r^{2} + b^{2}}

D

b2−r2r2−a2\sqrt{\frac{b^{2} - r^{2}}{r^{2} - a^{2}}}

Answer

b2−r2r2−a2\sqrt{\frac{b^{2} - r^{2}}{r^{2} - a^{2}}}

Explanation

Solution

The given circle is x2 + y2 = r2

Any tangent to the circle is y = mx ± r1+m2\sqrt{1 + m^{2}} if it is a tangent to the ellipse, then

r2(1+m2) = a2m2 + b2

∴ m = b2−r2r2−a2\sqrt{\frac{b^{2} - r^{2}}{r^{2} - a^{2}}}