what is the shortest distance between a parabola and the circle?

Question

what is the shortest distance between the parabola and the circle?

the equation of parabola is $$y^2=4ax$$

and the equation of circle is $$x^2+y^2-24y+81=0$$

if you can show graphically it will be more helpful!! thanks

Answer 1

HINT...find the general equation of the normal to the parabola at the point $P(at^2, 2at)$ and find the value of $t$ for which this normal passes through the centre of the circle. Then you can find the closest point on the parabola (with this value of $t$), and the rest is just considering distances and the radius of the circle.

Answer 2

The center of the circle lies at $C(0,12)$. If we take a point $P(b,\pm\sqrt{2ab})$ on the parabola, the tangent through $P$ has slope $\pm\sqrt{\frac{a}{2b}}$, while the line through $P$ and $C$ has slope $\frac{\pm\sqrt{2ab}-12}{b}$.

In order that $P$ is the point of minimum distance, the product of such slopes has to be $-1$.

So $b$, hence $P$, can be found by solving a third-degree equation.