Question

Find the relation between x and y where point (x, y) is equidistant from (2, -4) and (-2, 6)

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{A}\equiv \left(x,y\right),\mathrm{B}\equiv \left(2,-4\right)\mathrm{and}\mathrm{C}\equiv \left(-2,6\right). \\ \mathrm{Now},\mathrm{as}\mathrm{A}\mathrm{is}\mathrm{equidistan}\text{t}\mathrm{from}\mathrm{B}\mathrm{and}\mathrm{C}, \end{gathered}$$

$$\begin{gathered} \mathrm{AB}=\mathrm{AC} \\ \Rightarrow {\mathrm{AB}}^2={\mathrm{AC}}^2 \\ \Rightarrow {\left(x-2\right)}^2+{\left[y-\left(-4\right)\right]}^2={\left[x-\left(-2\right)\right]}^2+{\left(y-6\right)}^2 \\ \Rightarrow {\left(x-2\right)}^2+{\left(y+4\right)}^2={\left(x+2\right)}^2+{\left(y-6\right)}^2 \\ \Rightarrow x^2+4-4x+y^2+16+8y=x^2+4+4x+y^2+36-12y \\ \Rightarrow -4x+8y+20=4x-12y+40 \\ \Rightarrow -8x+20y=20 \\ \therefore -2x+5y=5 \\ \mathrm{This}\mathrm{is}\mathrm{the}\mathrm{required}\mathrm{relation}\mathrm{between}x\mathrm{and}y. \end{gathered}$$