JEE Mathematics 2023 Question with Solution

Given: $|z| = 1$ and the locus of $z + \frac{1}{z}$ is called $C_1$.

Find: The correct relation between the curves $C_1$ and $C_2$.

From the solution, the working explicitly concludes:

$$z = e^{i\theta} \implies z + \frac{1}{z} = e^{i\theta} + e^{-i\theta} = 2\cos\theta$$

Hence the locus of $z + \frac{1}{z}$ is a real line segment.

However, the provided the solution also contains a contradictory block mentioning

$$w = z + \frac{1}{z} = 4e^{i\theta} + \frac{1}{4}e^{-i\theta}$$

which corresponds to $|z| = 4$, not $|z| = 1$. Despite this inconsistency, the solution explicitly states The Correct Option is B.

Therefore, using the solution, the correct option is B.

Given: The question states $|z| = 1$.

Find: Which option must be selected based on the provided source.

If we use the question statement directly, then let

$$z = e^{i\theta}$$

so that

$$z + \frac{1}{z} = e^{i\theta} + e^{-i\theta} = 2\cos\theta$$

Thus $C_1$ is the line segment on the real axis from $-2$ to $2$.

But the options compare $C_1$ with $C_2$, and the definition of $C_2$ is missing from the question. Therefore the geometry cannot be completed from the question alone.

The source the solution is internally inconsistent: one part derives a result for $|z| = 4$ and says the curves intersect at $4$ points, while another part explicitly marks B as the correct option. Since the