Two tangent lines $l_1$ and $l_2$ are drawn from the point $\left(2,0\right)$ to the parabola $2y^2=-x$. If the lines $l_1$ and $l_2$ are also tangent to the circle ${\left(x-5\right)}^2+y^2=r$, then $17 r^2$ is equal to

Given, from point $\left(2,0\right)$ tangents are drawn on $2y^2=-x$, these tangents also touches the circle ${\left(x-5\right)}^2+y^2=r^2$, then value of $17r^2$ is:

Now equation of tangents when point is outside will be $y=mx-\frac{1}{8m}$, now $\left(2,0\right)$ will lie here so $m=\pm \frac{1}{4}$, putting the value of $m$ we get,

$x-4y=2$ and $4y+x=2$

Using distance of centre from tangent equal to radius, we get

$\left|\frac{5-0-2}{\sqrt{17}}\right|=r$ or $\left|\frac{5+0-2}{\sqrt{17}}\right|=r$

$\therefore r^2=\frac{9}{17}$

$\Rightarrow 17r^2=9$