Find the equation of the locus of points which are equidistant from the points $\left(2, 3\right)$ and $\left(5, 7\right)$.
Let the given points be $\;$ $A \left(2, 3\right)$ $\;$ and $\;$ $B \left(5, 7\right)$.
Let $P \left(x, y\right)$ be a point on the given locus.
As per question, $\;$ $PA = PB$
i.e. $\;$ $\left(PA\right)^2 = \left(PB\right)^2$
i.e. $\;$ $\left(x - 2\right)^2 + \left(y - 3\right)^2 = \left(x - 5\right)^2 + \left(y - 7\right)^2$
i.e. $\;$ $x^2 - 4x + 4 + y^2 - 6y + 9 = x^2 - 10x + 25 + y^2 - 14y + 49$
i.e. $\;$ $6x + 8y - 61 = 0$ $\;\;\;$ is the equation of the required locus.