JEE Main 2022 (Online) 26th June Morning ShiftQuestion 1
Let the common tangents to the curves $4\left(x^{2}+y^{2}\right)=9$ and $y^{2}=4x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of semi-minor and semi-major axes equal to $OQ$ and $6$, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac{l}{e^{2}}$ is equal to $\_\_\_\_\_\_$ .
Ellipse
Answer$4$
Solution
Tangent to $y^2=4x$: $y=mx+\frac1m$. Tangent to circle of radius $\frac32$: $\frac1{m^2}=\frac94(1+m^2)\Rightarrow 9m^4+9m^2-4=0\Rightarrow m^2=\frac13$. Tangents meet on the axis $y=0$: $x=-\frac1{m^2}=-3$, so $Q=(-3,0)$, $OQ=3$. Ellipse: $b=3,\ a=6$: $e^2=1-\frac9{36}=\frac34$, $l=\frac{2b^2}a=3$. $\frac l{e^2}=\frac{3}{3/4}=4$.