Cambridge AS & A Level. 2025. Mathematics (9709). May/June - Paper 3 (31) · Pure Mathematics 3 - Question 4

Find the complex numbers $z$ for which $\dfrac{z+5\mathrm{i}}{z-5}$ is real and $|z| = \sqrt{17}$. Give your answers in the form $z = x+\mathrm{i}y$, where $x$ and $y$ are real.