Question

Sol. [Key Concept: $|z - (4 + 8i)| = \sqrt{10}$ is a circle; the second condition is an ellipse with foci $(3,5),(5,11)$; the circle is concentric with radius equal to the ellipse's semi-minor axis, meeting it at $2$ points.]

$$|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5} $$So, $S_{1}(3,5)\&{\text{ }S}_{2}(5,11)$ are two foci of ellipse, where $2ae = S_{1}{\text{ }S}_{2} = 2\sqrt{10}\& 2a = 4\sqrt{5}$

$\Rightarrow a = 2\sqrt{5},e = \frac{1}{\sqrt{2}}\ \&\ b^{2} = a^{2}\left( 1 - e^{2} \right)$ gives $b^{2} = 10$

& center of an ellipse is mid point of both focus

∴ center $(4,8)$

Equation of circle is $|Z - (4 + 8i)| = \sqrt{10}$

⇒ Ellipse & given circle are concentric & radius of given circle is equal to length of minor axis of given ellipse

⇒ Two points which are common to both.