Maths Q2 – JEE-MAIN 2026

Home Maths Complex & QuadraticQuestion #81

The number of values of

z \in C

, satisfying the equations

| z - (4 + 8 i) | = \sqrt{10}

and

| z - (3 + 5 i) | + | z - (5 + 11 i) | = 4 \sqrt{5}

, is:

Clue 1 of 3: Geometric Interpretation

Recognize that the given equations represent geometric shapes in the complex plane. The first equation describes a circle, and the second describes an ellipse.

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Step 1: Identify Geometric Shapes and Parameters✦ Active

The first equation, $| z - (4 + 8 i) | = \sqrt{10}$ , represents a circle with center $C_{1} = (4, 8)$ and radius $R = \sqrt{10}$ .

The second equation, $| z - (3 + 5 i) | + | z - (5 + 11 i) | = 4 \sqrt{5}$ , represents an ellipse with foci $F_{1} = (3, 5)$ and $F_{2} = (5, 11)$ . The sum of the distances from any point on the ellipse to the foci is $2 a = 4 \sqrt{5}$ , so $a = 2 \sqrt{5}$ . The distance between the foci is $2 c = | F_{1} F_{2} | = \sqrt{(5 - 3)^{2} + (11 - 5)^{2}} = \sqrt{2^{2} + 6^{2}} = \sqrt{4 + 36} = \sqrt{40} = 2 \sqrt{10}$ , so $c = \sqrt{10}$ .

The center of the ellipse is the midpoint of the foci, $C_{E} = (\frac{3 + 5}{2}, \frac{5 + 11}{2}) = (4, 8)$ . The semi-minor axis length $b$ is calculated as $b = \sqrt{a^{2} - c^{2}} = \sqrt{(2 \sqrt{5})^{2} - (\sqrt{10})^{2}} = \sqrt{20 - 10} = \sqrt{10}$ .

Step 2: Analyze Concentricity and Distances○ Expand

Both the circle and the ellipse are centered at the same point $(4, 8)$ . The radius of the circle is $R = \sqrt{10}$ . For the ellipse, the semi-minor axis is $b = \sqrt{10}$ and the semi-major axis is $a = 2 \sqrt{5} = \sqrt{20}$ .

For any point $z$ on the ellipse, its distance from the center $(4, 8)$ ranges from $b$ (at the minor axis vertices) to $a$ (at the major axis vertices). Thus, $\sqrt{10} \leq | z - (4 + 8 i) | \leq \sqrt{20}$ .

💡 Teacher's Secret Hint

Note that the center of the circle and the ellipse are identical, simplifying the analysis.

Step 3: Determine Intersection Points○ Expand

For a point $z$ to satisfy both equations, it must lie on both the circle and the ellipse. This means its distance from the common center $(4, 8)$ must be exactly $\sqrt{10}$ . From the range of distances for the ellipse, $| z - (4 + 8 i) | = \sqrt{10}$ only occurs when $z$ is one of the minor axis vertices of the ellipse. An ellipse has exactly two minor axis vertices. Therefore, there are 2 values of $z$ that satisfy both equations.

💡 Teacher's Secret Hint

The minimum distance from the center to a point on the ellipse is its semi-minor axis length, $b$ . If this equals the circle's radius, these are the only intersection points.

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Maths Question 2 – JEE-MAIN 2026

Clue 1 of 3: Geometric Interpretation

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