Complex Numbers
Complex Numbers
star_batch_jee_advanced_2025
Grade None
Question:
If from a point $P$ representing the complex number $z_1$ on the curve $|z| = 2$, pair of tangents are drawn to the curve $|z|=1$, meeting at point $Q(z_2)$ and $R(z_3)$, then:
complex number $\frac{z_1 + z_2 + z_3}{3}$ will lie on the curve $|z| = 1$
$\left(\frac{4}{z_1} + \frac{1}{z_2} + \frac{1}{z_3}\right)\left(\frac{4}{z_1} - \frac{1}{z_2} - \frac{1}{z_3}\right) = 9$
$\arg\left(\frac{z_2}{z_3}\right) = \frac{2\pi}{3}$
orthocentre and circumcentre of $\triangle PQR$ will coincide
Step-by-Step Solution
Key Concept: Using inclusion-exclusion principle to count arrangements when certain objects must be excluded or constraints apply.
The total number of ways to select items is $5^3 - ^3C_1 \cdot 2^5 + ^3C_2 = 150$. Option (A) uses $^5P_3 = 60$. Option (B) counts parallelograms as $^5C_2 \times ^5C_2 = 150$. Option (C) gives $3^5 - ^3C_1 \cdot 2^5 + ^3C_2 = 150$ using inclusion-exclusion. Option (D) restates the same counting as $3^5 - ^3C_1 \cdot 2^5 + ^3C_2 = 150$.
Correct Answer: 1,2,3,4