JEE Main 2026 — 8 April, Evening Shift — Question 22

$|z-(3+5 i)|+|z-(5+11 i)|=4 \sqrt{5}$ So, $\mathrm{S}_{1}(3,5)$ & $\mathrm{S}_{2}(5,11)$ are two foci of ellipse, where $2 \mathrm{ae}=\mathrm{S}_{1} \mathrm{~S}_{2}=2 \sqrt{10} \& 2 \mathrm{a}=4 \sqrt{5}$ $\Rightarrow a=2 \sqrt{5}, e=\frac{1}{\sqrt{2}} \& b^{2}=a^{2}\left(1-e^{2}\right)$ gives $b^{2}=10$ & center of an ellipse is mid point of both focus ∴ center $(4,8)$ Equation of circle is $|\mathrm{Z}-(4+8 \mathrm{i})|=\sqrt{10}$ ⇒ 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.