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

`y=mx+sqrt(a^(2)m^(2)+b^(2))` is tangent to the ellipse. Equation of concentric circle be `x^(2)+y^(2)=r^(2)`
`y =mx pm rsqrt(1+m^(2))` ia tangent to the circle.
`therefore rsqrt(1+m^(2))=sqrt(a^(2)m^(2)+b^(2))`
`(1+m^(2))r^(2)=a^(2)m^(2)+b^(2)`
`m^(2)(r^(2)-a^(2))=b^(2)-r^(2)`
`therefore m=sqrt((r^(2)-b^(2))/(a^(2)-r^(2))`