JEE Mathematics 2023 Question with Solution

Given: $z_1 = 2 + 3i$, $z_2 = 3 + 4i$, and

$$|z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2$$

Find: The geometric nature of the set $S$.

Let

$$z = x + yi$$

Then

$$|z - z_1|^2 = (x-2)^2 + (y-3)^2$$

and

$$|z - z_2|^2 = (x-3)^2 + (y-4)^2$$

Also,

$$|z_1-z_2|^2 = |(2+3i)-(3+4i)|^2 = |-1-i|^2 = 1^2+1^2 = 2$$

Substituting into the given condition,

$$((x - 2)^2 + (y - 3)^2) - ((x - 3)^2 + (y - 4)^2) = 2$$

Expanding,

$$(x^2 - 4x + 4 + y^2 - 6y + 9) - (x^2 - 6x + 9 + y^2 - 8y + 16) = 2$$

After cancellation,

$$2x + 2y - 12 = 2$$

So,

$$2x + 2y = 14$$

which gives

$$x + y = 7$$

This is a straight line. Its intercepts on the coordinate axes are $7$ and $7$, so their sum is

$$7 + 7 = 14$$

Therefore, the correct option is A.

The solution labels option D, but the working clearly gives the line $x+y=7$, so the mathematically correct answer is A.

The final equation of the locus is

$$x+y=7$$

A first-degree equation in $x$ and $y$ always represents a straight line in the Argand plane. Setting $y=0$ gives the $x$-intercept $7$, and setting $x=0$ gives the $y$-intercept $7$. Hence the sum of intercepts is $14$.