JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

For all $z \in \mathbb{C}$ on the curve $C_1 : |z| = 4$ , let the locus of the point $z + \frac{1}{z}$ be the curve $C_2$ . Then:

Choose the correct answer:

Explanation

Solution: $C_1$ ek circle hai jiski radius $r=4$ hai.

Locus $w = z + \frac{1}{z}$ ke liye $z = 4e^{i\theta}$ rakhein:

w = 4e^{i\theta} + \frac{1}{4}e^{-i\theta} = \left(4 + \frac{1}{4}\right)\cos \theta + i\left(4 - \frac{1}{4}\right)\sin \theta

x = \frac{17}{4}\cos \theta, \quad y = \frac{15}{4}\sin \theta

Yeh ek ellipse ( $C_2$ ) hai jiska semi-major axis $a = 4.25$ aur semi-minor axis $b = 3.75$ hai.

Kyunki circle ki radius $4$ hai ( $3.75 < 4 < 4.25$ ), circle aur ellipse ek dusre ko 4 points par intersect karenge.

Sahi Option: (3)

Choose the correct answer:

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