JEE Main 2025 Apr 2 Shift 1, Mathematics Q10: Involving Two or More Circles

JEE Main2025Apr 2, Shift 1Mathematics

Let $z$ be a complex number such that $∣ z ∣ = 1$ . If $\frac{2 + k ^{2} z}{k + z ˉ} = k z$ , $k \in R$ , then the maximum distance of $k + i k^{2}$ from the circle $∣ z - (1 + 2 i) ∣ = 1$ is:

A
$5 + 1$
B
2
C
3
D
$3 + 1$

Solution

Solve for k. Cross-multiply:

$2 + k^{2} z = k z (k + \overset{z}{ˉ}) = k^{2} z + k z \overset{z}{ˉ}$

Since $∣ z ∣ = 1$ , $z \overset{z}{ˉ} = 1$ :

$2 + k^{2} z = k^{2} z + k \Rightarrow k = 2$

Geometric setup. The point is $k + i k^{2} = 2 + 4 i$ , i.e. $P (2, 4)$ . The circle has centre $C (1, 2)$ and radius $r = 1$ .

Distance from $P$ to the centre:

$∣ P C ∣ = (2 - 1)^{2} + (4 - 2)^{2} = 1 + 4 = 5$

Maximum distance from $P$ to any point on the circle is $∣ P C ∣ + r$ :

$d_{m a x} = 5 + 1$

$∴$ Option (1) is correct.

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JEE Main 2025 Apr 2 Shift 1, Mathematics Q10: Involving Two or More Circles

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