Complex Numbers
Complex Numbers
Allen Star Batch
Grade 11
Question:
If the imaginary part of $\frac{z-3}{e^{i\theta}} + \frac{e^{i\theta}}{z-3}$ is zero, then $z$ can lie on
a circle with unit radius
a circle with radius $3$ units
a straight line through the point $(3, 0)$
a parabola with the vertex $(3, 0)$
Step-by-Step Solution
Key Concept: Stars and bars combined with the hockey stick identity converts the counting problem into a single binomial coefficient.
Let $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ where $x, y, z$ are positive components. Given $|\vec{r}| \leq 12$, we have $x + y + z \leq 12$. The number of positive integral solutions is $\sum_{n=3}^{12} ^{n-1}C_2 = ^{2}C_2 + ^{3}C_2 + \cdots + ^{11}C_2$. This equals $^{12}C_3$ by the hockey stick identity, giving $I = ^{12}C_3 = 220$.
Correct Answer: 1,3