Why are there only two answers for this co-ordinate geometry question? A co-ordinate geometry question reads "Find value of $p$ such that area of triangle $A (p,2p), B(-1,6), C (3,1)$ is $10$ sq. units. Only 2 values come for $p$. Why does that happen? Shouldn't there be infinite values of $p$ because given two points we can find infinite points forming a triangle with same area. Edit: How to prove that there will be only 2 points?

Let $h$ be the altitude of the triangle to the base $\overline{BC}$. Since $BC=\sqrt{41}$, then we must have $h = \dfrac{20}{\sqrt{41}}$. The point $A = (p,2p)$ lies on the line $y=2x$, and there can only be two points on that line that are a distance of $h$ from the line $\overleftrightarrow{BC}$.

Computationally, the equation for the distance of a point $(x_0,y_0)$ from the line $ax+by+c=0$ is $d = \dfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$. Since the equation of the line $\overleftrightarrow{BC}$ is $5x+4y-19=0$, then you need to solve the equation

$$\dfrac{20}{\sqrt{41}} = \dfrac{|5p+4(2p)-19|}{\sqrt{41}} $$

There will be two solutions.