Let z be a complex number. Assertion (A): The equation |z - 1| + |z - i| = 2 represent an ellipse. Reason (R): The equation |z -z1| + |z - z2|= k, where k is positive, represent an ellipse.

Question

Let z be a complex number. Assertion (A): The equation |z - 1| + |z - i| = 2 represent an ellipse. Reason (R): The equation |z -z1| + |z - z2|= 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.

Tardigrade Admin · Accepted Answer

Reason is true only if k > |z1 - z2| Here k = 2, z1 = 1, z2 = i and |z1 - z2|= |1 -i|= √2 ⇒ 2 > |1 -i|= √2 ⇒ Reason (R) is false but Assertion (A) is true

Q. 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.

Both (A) & (R) are individually true & (R) is correct explanation of (A).

Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

(A) is true but (R) is false.

(A) is false but (R) is true.

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

Q. Let z be a complex number. Assertion (A) : The equation ∣z−1∣+∣z−i∣=2 represent an ellipse. Reason (R) : The equation ∣z−z1​∣+∣z−z2​∣=k, where k is positive, represent an ellipse.