Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
Equation of line through $a$ and $ib$ such that $a, b \in \mathbb{R}$ and $a, b \neq 0$ is
$z\left(\frac{1}{a} + \frac{1}{2ib}\right) + \bar{z}\left(\frac{1}{2a} - \frac{1}{2ib}\right) = 1$
$z\left(\frac{1}{2a} + \frac{i}{2b}\right) + \bar{z}\left(\frac{1}{2a} + \frac{i}{2b}\right) = 1$
$i\left(\frac{1}{2a} - \frac{i}{2b}\right) + \bar{z}\left(\frac{1}{2a} - \frac{i}{2b}\right) = 1$
$z\left(\frac{1}{2a} - \frac{i}{2b}\right) + \bar{z}\left(\frac{1}{2a} - \frac{i}{2b}\right) = 1$
Step-by-Step Solution
Key Concept: Divide by the number of repetitions (2!) when groups are indistinguishable to avoid overcounting.
The required number of teams is $\frac{^{10}C_4 \cdot ^6C_3 \cdot ^3C_3}{2!} = \frac{10!}{4! \cdot 6!} \times \frac{6!}{3! \cdot 3!} \times 3 \times \frac{1}{^{10}C_4 \times ^5C_2} = 2100$.
Correct Answer: 1,4