Question

Let $z$ be a complex number.
Assertion (A) : The equation $|z - 1| + |z - i| = 2$ represent an ellipse.
Reason (R) : The equation $|z -z_1| + |z - z_2|= k$, where $k$ is positive, represent an ellipse.

• 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)

Reason is true only if $k > |z_1 - z_2|$
Here $k = 2, z_1 = 1, z_2 = i$
and $|z_1 - z_2|= |1 -i|= \sqrt{2}$
$\Rightarrow 2 > |1 -i|= \sqrt{2}$
$\Rightarrow $ Reason (R) is false but Assertion (A) is true