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  • The slope of a common tangent to the ellipse <img alt="\mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7Bx%5E2%7D%7Ba%5E2%7D&am

The slope of a common tangent to the ellipse \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 }and a concentric circle of radius r is

Option: 1

\mathrm{\tan ^{-1} \sqrt{\frac{r^2-b^2}{a^2-r^2}}}


Option: 2

\mathrm{ \sqrt{\frac{r^2-b^2}{a^2-r^2}}}


Option: 3

\mathrm{{\frac{r^2-b^2}{a^2-r^2}}}


Option: 4

\mathrm{\sqrt{\frac{a^2-r^2}{r^2-b^2}}}


Answers (1)

best_answer

\mathrm{\text { We have, } \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \text { and } x^2+y^2=r^2}

Equation of tangent to circle is \mathrm{x \cos \theta+y \sin \theta=r}
and equation of tangent to ellipse is

\mathrm{y=m x \pm \sqrt{a^2 m^2+b^2}}

If it is tangent to circle, then ⊥ distance from (0, 0) is equal to r.

\mathrm{\begin{aligned} & \Rightarrow\left|\frac{\sqrt{a^2 m^2+b^2}}{\sqrt{m^2+1}}\right|=r \\ & \Rightarrow a^2 m^2+b^2=m^2 r^2+r^2 \Rightarrow m=\sqrt{\frac{r^2-b^2}{a^2-r^2}} \end{aligned}}

 

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