<s>[INST] Finding perpendicular bisector of the line segement joining $ (-1,4)\;\text{and}\;(3,-2)$ 
Find the perpendicular bisector of the line joining the points $(-1,4)\;\text{and}\;(3,-2).\;$

I know this is a very easy question, and the answer is an equation. So any hints would be very nice. thanks
 [/INST] If the point $P=(x,y)$ lies on the perpendicular bisector of the points $A=(-1,4)$ and 
$B= (3, -2)$, then the distances $PA$ and $PB$ must be the same.
Then
$$
PA^2 = PB^2  \\
\Rightarrow (x+1)^2 + (y-4)^2 = (x-3)^2 + (y+2)^2 \\
\Rightarrow  x^2 +2x + 1 + y^2 -8y +16 = x^2 -6x +9 +y^2 + 4y + 4 \\
\Rightarrow 2x - 8y +17 = -6x +4y +13 \\
\Rightarrow 8x - 12y + 4 = 0 \\
\Rightarrow 2x -3y + 1 = 0
$$
So, the perpendicular bisector has equation $2x -3y + 1 = 0$.
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