If the common tangents to the parabola x^2=4y and the circle x^2+y^2=4 intersect at the po…

Question

If the common tangents to the parabola $x^2=4y$ and the circle $x^2+y^2=4$ intersect at the point $P$, then the distance of $P$ from the origin, is:

Accepted Answer

$x^2=4y$; tangent at parameter $m$ (slope): $y=mx-m^2$. Tangent to the circle $x^2+y^2=4$:
$$\frac{|{-m^2}|}{\sqrt{m^2+1}}=2\Rightarrow m^4=4m^2+4\Rightarrow m^2=2+2\sqrt2\ (\text{taking the positive root}).$$

By symmetry the tangents (slopes $\pm m$) meet on the $y$-axis at $(0,-m^2)$:

$$OP=m^2=2+2\sqrt2=2(\sqrt2+1).$$

Solutions · 1