Question

If the tangent of slope $m$ at a point of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through $(2a,0)$ and if $e$ is the eccentricity of the ellipse, then

• A. $m^2+e^2=1$
• B. $2m^2+e^2=1$
• C. $3m^2+e^2=1$
• D. $3m^2=1+e^2$

Answer 1

The correct option is C $3m^2+e^2=1$

$y=mx\pm \sqrt{a^2{m}^2+b^2}$ is equation of any tangent to the given ellipse with slope $m$.
Since it passes through $(2a,0)$,
$0=2am\pm \sqrt{a^2{m}^2+b^2}$
$\Rightarrow 4a^2{m}^2=a^2{m}^2+b^2$
$\Rightarrow 3a^2{m}^2=a^2(1-e^2)$
$\Rightarrow 3m^2+e^2=1$
Hence, option 'C' is correct.