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

Question

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.

Answer 1

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 $y=-x$.

Answer 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?

Answer 3

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:

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