Complex Numbers
Complex Numbers
star_batch_jee_advanced_2025
Grade None
Question:
If $\frac{z-z_1}{z-z_2}=3$, where $z_1$ and $z_2$ are fixed complex numbers and $z$ is a variable complex number, then $z$ lies on a :
Circle with $z_1$ as its interior point
Circle with $z_2$ as its interior point
Circle with $z_1$ and $z_2$ as its interior points
Circle with $z_1$ and $z_2$ as its exterior points
Step-by-Step Solution
Key Concept: The equation $|z-z_1|=3|z-z_2|$ represents an Apollonius circle where the locus is equidistant (in ratio 3:1) from two fixed points, with $z_2$ lying inside since it's the closer reference point.
Given $\frac{z-z_1}{z-z_2}=3$, we interpret this as $|z-z_1|=3|z-z_2|$. This means the distance from $z$ to $z_1$ is always 3 times the distance from $z$ to $z_2$. By the Apollonius circle theorem, the locus of points whose distances to two fixed points are in a constant ratio (not equal to 1) forms a circle. Since the ratio is 3:1, the circle is closer to $z_2$ than to $z_1$. To determine which point is interior, note that points between $z_1$ and $z_2$ satisfy the condition, and since $z_2$ is closer to the locus, $z_2$ must be an interior point of the circle.
Correct Answer: 2