Respuesta :
Given the points : ( 3 , 8 ) and ( -2 , 5 )
We need to find the coordinates of the point equidistant from the given point and should be on the y - axis
as the point will be on y - axis : the x - coordinates of the point will be 0
So, let the coordinates of the point is ( 0 , a )
The distance between any two points is given by the formula :
[tex]d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]So, the distance between ( 3 , 8 ) and ( 0 , a ) will be :
[tex]d_1=\sqrt[]{(0-3)^2+(a-8)^2}=\sqrt[]{9+(a-8)^2}[/tex]And the distance between ( -2 , 5 ) and ( 0 , a ) will be :
[tex]d_2=\sqrt[]{(0+2)^2+(a-5)^2}=\sqrt[]{4+(a-5)^2}[/tex]the two distances are equal
[tex]\begin{gathered} d_1=d_2 \\ \sqrt[]{9+(a-8)^2}=\sqrt[]{4+(a-5)^2} \end{gathered}[/tex]squaring both sides then solve for a:
[tex]\begin{gathered} 9+(a-8)^2=4+(a-5)^2 \\ 9+a^2-16a+64=4+a^2-10a+25 \\ -16a+73=-10a+29 \\ -16a+10a=29-73 \\ -6a=-44 \\ \\ a=\frac{-44}{-6}=\frac{22}{3}=7\frac{1}{3} \end{gathered}[/tex]so, the answer is the coordinates of the point is ( 0 , 7 1/3 )
