If $\mathrm{f} : \mathrm{R}\to \mathrm{N}\cup \left\{0\right\}$, where f(area of triangle joining points P(5, 0), Q(8,4) and R(x, y) such that angle PRQ is a right angle) = number of triangles, then which of the following is true?

Since $\mathrm{\angle }\mathrm{PRQ}=\mathrm{\pi }/2$ and points P, Q, R lie on the circle with PQ as diameter.

Also, PQ = 5.

Now, the maximum area of the triangle is

${\mathrm{\Delta }}_{max}=\frac{1}{2}\left(5\right)\left(\frac{5}{2}\right)=6.25$

For area = 5, we have four symmetrical positions of point R(shown as R₁, R₂, R₃, R₄).

For area = 6.25, we have exactly two points.

For area = 7, no such points exist.