# Checking If a Point Is On the Parabola Given the Focus and Directrix

The question is as follows:

Is the point $P = (4,2)$ on the parabola that has a focus $F = (0,4)$ and the directrix $y=-x$? Show your work.

Supposedly there is an easier way than to write out the equation of the parabola. How is that possible? Any help will be greatly appreciated.

It is enough to draw a picture:

The distance of $P$ from the $y=-x$ line is $3\sqrt{2}$, while the length of $PF$ is $2\sqrt{5}$.
It follows that $P$ does not lie on the parabola with focus at $F$ and directrix $y=-x$.

• Oh! I see what you can do! Thank you so much. It was so simple of a method and it didn't even cross my mind. Thanks again! Sep 23 '17 at 23:07

You start by saying that the distance between a point $(x,y)$ in the parabola and the focus is equal to the distance between the parabola and the directrix. For example, the distance between the parabola and the focus in your case is:

$$\sqrt{(x-4)^2+(y-2)^2}$$

Hint:

The parabola is the locus of points equidistant from the focus and the directrix.

Can you find the distance of the point $P$ from the focus and from the directrix?