The number of points z in the complex plane satisfying both the equations $|\mathrm{z}-4-8\mathrm{i}|=\sqrt{10}$ and $|\mathrm{z}-3-5\mathrm{i}|+|\mathrm{z}-5-11\mathrm{i}|=4\sqrt{5}\text{ is }$

1st eq is a circle with centre (4, 8) and radius = $\sqrt{10}$. 2nd eq is an ellipse whose foci are S(3,5),S'(5,11) . Centre C(4,8),a = 2$\sqrt{5}$,ae = $\sqrt{10}$ and hence b = $\sqrt{10}$. $\therefore$ 1st eq is circle on SS' as diameter and hence B(7, 7) and B'(1,9) are the common points.