Question

The values of y for which the point (2, –4) is equidistant from the points (3, 8) and (–10, y), are _________.

Answer 1

Let the point P(2, –4) is equidistant from the points A(3, 8) and B(–10, y).

Thus, AP = BP
$$\begin{gathered} \Rightarrow {\left(AP\right)}^2={\left(BP\right)}^2 \\ \Rightarrow {\left(\sqrt{{\left(2-3\right)}^2+{\left(-4-8\right)}^2}\right)}^2={\left(\sqrt{{\left(2-\left(-10\right)\right)}^2+{\left(-4-y\right)}^2}\right)}^2 \\ \Rightarrow {\left(-1\right)}^2+{\left(-12\right)}^2={\left(2+10\right)}^2+{\left(-4-y\right)}^2 \\ \Rightarrow 1+144={\left(12\right)}^2+{\left(4+y\right)}^2 \\ \Rightarrow 1+144=144+16+y^2+8y \\ \Rightarrow 1=16+y^2+8y \\ \Rightarrow y^2+8y+15=0 \\ \Rightarrow y^2+5y+3y+15=0 \\ \Rightarrow y\left(y+5\right)+3\left(y+5\right)=0 \\ \Rightarrow \left(y+3\right)\left(y+5\right)=0 \\ \Rightarrow y=-5,-3 \\ \mathrm{Therefore},y=-5,-3. \end{gathered}$$

Hence, the values of y for which the point (2, –4) is equidistant from the points (3, 8) and (–10, y), are –5 and ​–3.