Complex Numbers: For all zinmathbb C on the curve C_1: |z|=4 , let the locus of the… | JEE Main 2023

JEE Main 2023 (Online) 31st January Morning ShiftQuestion 1

For all $z\in\mathbb C$ on the curve $C_1: |z|=4$, let the locus of the point $z+\frac1z$ be the curve $C_2$. Then

Complex Numbers

the curves $C_1$ and $C_2$ intersect at $4$ points✓ Correct answerthe curve $C_1$ lies inside $C_2$the curves $C_1$ and $C_2$ intersect at $2$ pointsthe curve $C_2$ lies inside $C_1$

Solution

With $z=4e^{it}$, $z+\frac1z=\frac{17}4\cos t+i\frac{15}4\sin t$, an ellipse with semi-axes $\frac{17}4>4>\frac{15}4$, so it cuts the circle $|z|=4$ in $4$ points.

Complex Numbers - JEE Mathematics question (JEE Main 2023 (Online) 31st January Morning Shift)

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