JEE MAIN 2026 · Complex Numbers and Quadratic Equations

The first equation is a circle with centre $(4, 8)$ and radius $\sqrt{10}$.

The second is an ellipse with foci $S_1(3, 5)$ and $S_2(5, 11)$ and $2a = 4\sqrt{5}$, so $a = 2\sqrt{5}$.

• Its centre is the midpoint of the foci: $(4, 8)$, the same as the circle's centre.
• $2c = |S_1S_2| = \sqrt{2^2 + 6^2} = 2\sqrt{10}$, so $c = \sqrt{10}$.
• $b^2 = a^2 - c^2 = 20 - 10 = 10$, so $b = \sqrt{10}$.

The circle has radius equal to the semi-minor axis and the same centre, so it lies inside the ellipse and touches it only at the two ends of the minor axis.

Answer: 2 values of $z$.