Complex Numbers
Real Part Conditions — System of Equations in Complex Variables
nta_pyq_2023_apr
Grade 11
Question:
Let $a\neq b$ be two non-zero real numbers. Then the number of elements in the set $X=\{z\in\mathbb{C}:\ \text{Re}(az^2+bz)=a\ \text{and}\ \text{Re}(bz^2+az)=b\}$ is equal to
Step-by-Step Solution
Key Concept: Write $z=x+iy$. Extract real parts from both conditions to get two equations in $x,y$. Subtracting gives $x^2-y^2-x=1$; adding gives $x^2-y^2+x-1=0$.
$x=0,\ -y^2=1\Rightarrow$ no solution. Number of elements $=0$.
Correct Answer: 1