If the point (x,y) is equdistant from the points A(5,1) and B(1,5) prove that 3x=2y

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Please find below the solution to the asked query:

We have two points  A( 5 , 1 ) , B(  1 , 5 ) 


We know distance formula  :  d = $\sqrt{{\left(x_2 - x_1\right)}^2 + {\left(y_2 - y_1\right)}^2}$ ,

Given : If the point (x , y ) is equidistant from the points A( 5 , 1 ) and B( 1 , 5 )  , So


$$\begin{gathered} \sqrt{{\left(5- x\right)}^2 + {\left( 1 - y\right)}^2} = \sqrt{{\left( x- 1\right)}^2 + {\left( y - 5\right)}^2} \\ \mathrm{Squaring} \mathrm{both} \mathrm{side} \mathrm{we} \mathrm{get} : \\ \Rightarrow {\left(5- x\right)}^2 + {\left( 1 - y\right)}^2= {\left( x- 1\right)}^2 + {\left( y - 5\right)}^2 \\ \mathrm{We} \mathrm{know} : {\left(\mathrm{a} - \mathrm{b} \right)}^2 = {\mathrm{a}}^2 + {\mathrm{b}}^2 - 2 \mathrm{a} \mathrm{b} , \mathrm{So} \\ \Rightarrow 25 + x^2 - 10 x + 1 + y^2 - 2 y= x^2 + 1 - 2x + y^2 + 25 - 10 y \\ \Rightarrow - 10 x - 2 y= - 2x - 10 y \\ \Rightarrow - 8 x =- 8 y \\ \mathbf{\Rightarrow }\mathit{x}\mathbf{ }\mathbf{ }\mathbf{=}\mathit{y}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{\hspace{0.17em}}\mathit{A}\mathit{n}\mathit{s}\mathbf{ }\mathbf{)} \end{gathered}$$

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