JEE Main 2022 (Online) 28th July Evening ShiftQuestion 1
Two tangent lines $l_1$ and $l_2$ are drawn from the point $(2,0)$ to the parabola $2y^2=-x$. If the lines $l_1$ and $l_2$ are also tangent to the circle $(x-5)^2+y^2=r$, then $17r$ is equal to ________.
Parabola
Answer$9$
Solution
$2y^2=-x$ is $y^2=-\frac x2$. Tangent $y=mx+c$ with $c=\frac{a}{m}$, $a=-\frac18$, so $c=-\frac1{8m}$. Through $(2,0)$: $2m=\frac1{8m}\Rightarrow m=\pm\frac14$. Tangents: $x\mp4y-2=0$. Tangent to circle with center $(5,0)$: $r=\left(\frac{|5-2|}{\sqrt{17}}\right)^2=\frac9{17}$. $17r=9$.