Question

Let $z$ be a complex number
Assertion (A) :The equation $|z - 1| - |z - i| = 1$ represents a hyperbola.
Reason (R) : The equation $|z - z_1| - |z - z_2| = k$ is a hyperbola.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

Answer: (C)

The equation ||$z-z_1|-| z-z_2||=k$ is a hyperbola only
if $k<\left|z_1-z_2\right|$
Here $k=1 \&\left|z_1-z_2\right|=\sqrt{2} \nless 1\left\{z_1=1, z_2=i\right\}$
Reason $( R )$ is false & Assertion $( A )$ is true.