Question

If the distance of $p(x,y)$ from $a(5,1)$ and $b(-1,5)$ are equal, prove that $3x=2y$.

Answer 1

Prove the statement using distance formula

Distance between two points $(x_1,y_1)and(x_2,y_2)$ is calculated by:

$D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Given: Point $p$ is equidistance from point $a$ and $b$.

Therefore, $pa=pb$

$$\begin{gathered} \Rightarrow pa=pb \\ \Rightarrow \sqrt{{\left(5-x\right)}^2+{\left(1-y\right)}^2}=\sqrt{{\left(-1-x\right)}^2+{\left(5-y\right)}^2} \\ \Rightarrow 25+x^2-10x+1+y^2-2y=1+x^2+2x+25+y^2-10y \\ \Rightarrow 8y=12x \\ \Rightarrow 3x=2y \end{gathered}$$

Hence, proved that $3x=2y$.