Find the points on the locus of points that are 3 units from X axis and 5 units from the point $(5,1)$.
Let $P \left(h,k\right)$ be a point on the required locus.
$P$ is 3 units from the X axis.
$\implies$ Equation of locus is: $\hspace{1em}$ $k = \pm 3$ $\;\;\; \cdots \; (1)$
Also, $P$ is 5 units from the point $(5,1)$.
i.e. $\sqrt{\left(h - 5\right)^2 + \left(k - 1\right)^2} = 5$
$\therefore$ $\;$ Equation of locus is: $\hspace{1em}$ $\left(h - 5\right)^2 + \left(k -1\right)^2 = 5^2$ $\;\;\; \cdots \; (2)$
Substituting $k = + 3$ in equation $(2)$ gives
$\left(h - 5\right)^2 + \left(3 - 1\right)^2 = 25$
i.e. $h^2 - 10 h + 25 + 4 = 25$
i.e. $h^2 - 10 h + 4 = 0$
i.e. $h = \dfrac{10 \pm \sqrt{100 - 16}}{2} = 5 \pm \sqrt{21}$
Substituting $k = - 3$ in equation $(2)$ gives
$h^2 - 10 h + 25 + \left(-3 - 1\right)^2 = 25$
i.e. $h^2 - 10 h + 16 = 0$
i.e. $\left(h - 2\right) \left(h - 8\right) = 0$
$\implies$ $h = 2$, $h = 8$
$\therefore$ $\;$ The points on the required locus are: $\hspace{1em}$ $\left(5 + \sqrt{21}, 3\right)$, $\left(5 - \sqrt{21}, 3\right)$, $\left(2, -3\right)$ and $\left(8, -3\right)$