Complex Numbers
Circle from Re(w)=0 and Chord Length from Im(w)=0
nta_pyq_2023_apr
Grade 11
Question:
Let $w=z\bar{z}+k_1z+k_2iz+\lambda(1+i),\ k_1,k_2\in\mathbb{R}$. Let $\text{Re}(w)=0$ be the circle $C$ of radius 1 in the first quadrant touching the line $y=1$ and the $y$-axis. If the curve $\text{Im}(w)=0$ intersects $C$ at $A$ and $B$, then $30(AB)^2$ is equal to _______.
Step-by-Step Solution
Key Concept: $\text{Re}(w)=0$ gives $x^2+y^2+k_1x-k_2y+\lambda=0$. Circle touching $y=1$ and $y$-axis in Q1 with radius 1 has centre $(1,2)$: equation $x^2+y^2-2x-4y+4=0$.
$A=(0,2),\ B=(\frac{2}{5},\frac{14}{5})$. $(AB)^2=\frac{4}{25}+\frac{16}{25}=\frac{4}{5}$. $30(AB)^2=24$.
Correct Answer: 24