Find the slope of a common tangent to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` and a concentric circle of radius `rdot`

Options

A. `tan^(-1) sqrt((r^(2) - b^(2))/(a^(2) - r^(2)))`
B. `sqrt((r^(2) - b^(2))/(a^(2) - r^(2)))`
C. `(r^(2) - b^(2))/(a^(2) - r^(2))`
D. `sqrt((a^(2) - r^(2))/(r^(2) - b^(2)))`

Answer

B

Explanation

The equation of any tangent to the given ellipse is
`y = mx + sqrt(a^(2)m^(2) + b^(2))`
If it touches `x^(2) + y^(2) = r^(2).` Then,
`sqrt(a^(2)m^(2)b^(2)) = rsqrt(1 + m^(2)) " " ["Using " c = a sqrt(1 + m^(2))]`
`rArr m^(2)(a^(2) - r^(2)) = r^(2) - b^(2) rArr m = sqrt((r^(2) - b^(2))/(a^(2) - r^(2)))`