Complex Numbers
Complex Numbers
star_batch_jee_advanced_2025
Grade 11
Question:
If $|z| = 2$ and $\frac{z_1 - z_3}{z_2 - z_3} = \frac{z - 2}{z + 2}$, then $z_1, z_2, z_3$ will be vertices of a/an:
equilateral triangle
acute angled triangle
right angled triangle
None of the above
Step-by-Step Solution
Key Concept: Use multinomial coefficients to count arrangements within each case, then multiply by the number of ways to distribute distinct items.
The problem involves distributing objects into categories with constraints. Three cases are analyzed based on values of $P_1$, $P_2$, $P_3$: Case 1 gives $\frac{8!}{1!3!4!} \times 13 = 1680$, Case 2 gives $\frac{8!}{2!3!3!} \times \frac{13}{2} = 1680$, and Case 3 gives $\frac{8!}{2!2!4!} \times \frac{13}{2} = 1260$. The total is $1680 + 1680 + 1260 = 4620$. Therefore $K \times P_3 = 4620$, so $K = \frac{4620}{210} = 22$.
Correct Answer: 3