Let $\mathrm{z}_{1}=2+3 \mathrm{i}$ and $\mathrm{z}_{2}=3+4 \mathrm{i}$. The set $S=\left\{\mathrm{z} \in \mathrm{C}:\left|\mathrm{z}-\mathrm{z}_{1}\right|^{2}-\left|\mathrm{z}-\mathrm{z}_{2}\right|^{2}=\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|^{2}\right\}$ represents a

(A) straight line with sum of its intercepts on the coordinate axes equals 14

(B) hyperbola with the length of the transverse axis 7

(C) straight line with the sum of its intercepts on the coordinate axes equals -18

(D) hyperbola with eccentricity 2

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Complex Numbers

Let

z_{1} = 2 + 3 i

and

z_{2} = 3 + 4 i

. The set

S = {z \in C : ∣ z - z_{1} ∣^{2} - ∣ z - z_{2} ∣^{2} = ∣ z_{1} - z_{2} ∣^{2}}

represents a

(A)straight line with sum of its intercepts on the coordinate axes equals 14✓ Correct
(B)hyperbola with the length of the transverse axis 7
(C)straight line with the sum of its intercepts on the coordinate axes equals -18
(D)hyperbola with eccentricity 2

Answer: (A)

Writing

z = x + i y

∣ z - z_{1} ∣^{2} - ∣ z - z_{2} ∣^{2} = (x - 2)^{2} + (y - 3)^{2} - (x - 3)^{2} - (y - 4)^{2} = 2 x + 2 y - 12

, while

∣ z_{1} - z_{2} ∣^{2} = ∣ - 1 - i ∣^{2} = 2

. The condition becomes

2 x + 2 y - 12 = 2

, i.e.

x + y = 7

. This is a straight line cutting the axes at

(7, 0)

and

(0, 7)

, so the sum of its intercepts is

14

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Single correct option