Two tangent lines l_1 and l_2 are drawn from the point (2,0) to the parabola 2y^2=-x.

Question

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 \_\_\_\_\_\_\_\_.

Answer: 9

Accepted Answer

$y^2=-\frac x2$ has the form $y^2=4Ax$ with $A=-\frac18$; tangent: $y=mx+\dfrac A m=mx-\dfrac1{8m}$.

Through $(2,0)$: $2m=\dfrac1{8m}\Rightarrow m^2=\dfrac1{16}\Rightarrow m=\pm\dfrac14$, giving lines $x-4y-2=0$ and $x+4y-2=0$.

Distance from $(5,0)$: $\dfrac{|5-2|}{\sqrt{17}}=\dfrac3{\sqrt{17}}=\sqrt r\Rightarrow r=\dfrac9{17}$.

$17r=9.$

Solutions · 1