Let z_1 = 2 + 3i and z_2 = 3 + 4i. The set S = \ z ∈ C : |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2 \ represents a

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

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

hyperbola with the length of the transverse axis 7

Let z_1 = 2 + 3i and z_2 = 3 + 4i. The set S = \ z ∈ C : |z… — JEE Main 2023 Mathematics PYQ

JEE Main 2023 · 25 Jan · Shift 1

Let

z_1 = 2 + 3i

and

z_2 = 3 + 4i

. The set

S = \left\{ z \in C : |z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2 \right\}

represents a

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

Show answer and solution

Correct option: A

Solution

Let z = x + iy. Then:

|z - z₁|² = (x-2)² + (y-3)²
|z - z₂|² = (x-3)² + (y-4)²
|z₁ - z₂|² = 1 + 1 = 2

Expanding the condition:
(x-2)² + (y-3)² - (x-3)² - (y-4)² = 2

(x²-4x+4) - (x²-6x+9) + (y²-6y+9) - (y²-8y+16) = 2

2x - 5 + 2y - 7 = 2

2x + 2y = 14

x + y = 7

This is a straight line. The x-intercept is 7 (set y=0) and y-intercept is 7 (set x=0), so the sum of intercepts = 7 + 7 = 14.

[JEE Main 2023 · 25 Jan · Shift 1]

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