Question

A tangent to $3x^2+4y^2=12$ is equally inclined with the coordinate axes. If $d$ is the perpendicular distance from the centre of the ellipse to this tangent, then $2d^2=$ units

Answer 1

Equation of the ellipse can be written as $\frac{x^2}{4}+\frac{y^2}{3}=1$
Since tangent makes equal angles with coordinate axes,
$\therefore$ Equation of the tangent is $y=\pm x+c$
Now, using tangency condition,
$c^2=a^2{m}^2+b^2=7\Rightarrow c=\pm \sqrt{7}$

Hence distance from $(0,0)$ i.e., centre of ellipse is
$d=\left|\frac{\sqrt{7}}{\sqrt{1+1}}\right|=\sqrt{\frac{7}{2}}$ units
$\therefore 2d^2=7$