Let z₁ = 2+3i and z₂ = 3+4i. The set S = {z ∈ ℂ : |z-z₁|² - |z-z₂|² = |z₁-z₂|²} represents a (1) str

Q: Wait, why do we reject the negative root here? Or why can't we just use energy conservation?

That is a great question! In this problem, we are not dealing with energy conservation, but rather with the geometric locus of a point. The equation $|z-z_1|^2 - |z-z_2|^2 = |z_1-z_2|^2$ is a direct algebraic constraint. When we expand it, the quadratic terms $x^2$ and $y^2$ cancel out naturally, leaving a linear equation. There is no 'negative root' to reject because the equation is linear, not quadratic.

Q: I got confused with the signs when subtracting the two expressions. Is there a trick?

The trick is to be methodical. Always expand each term fully within parentheses before performing the subtraction. For example, write the first expression in one set of brackets and the second in another, then distribute the negative sign to every term inside the second bracket. This prevents the common mistake of missing a sign change.

Q: Do I really need to convert to $z = x+iy$? Is there a purely complex number way?

You absolutely can use the property $|w|^2 = w \bar{w}$. It is elegant and often faster for those comfortable with complex algebra. However, for JEE, the Cartesian method ($z = x+iy$) is often safer and less prone to algebraic errors, as it reduces the problem to standard coordinate geometry, which is a very powerful tool in your arsenal.

Question

Let $z_1 = 2+3i$ and $z_2 = 3+4i$. The set $S = \{z \in \mathbb{C} : |z-z_1|^2 - |z-z_2|^2 = |z_1-z_2|^2\}$ represents a

Options:

(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

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler, to the fascinating world of complex numbers. Often, students see $z = x+iy$ and think only of imaginary numbers, but today, we are going to see them for what they truly are: a map of reality.

We are tasked with finding the locus of a point $z$ that satisfies the condition $|z-z_1|^2 - |z-z_2|^2 = |z_1-z_2|^2$, where $z_1 = 2+3i$ and $z_2 = 3+4i$.

Imagine you are standing on the Argand plane. You have two fixed points, $z_1$ and $z_2$. You are looking for a path—a locus—of a moving point $z$ that maintains a very specific relationship between its distances to these two fixed points.

### The Algebraic Grind

To solve this, we must translate the complex language into the language of coordinates. Let $z = x+iy$.

The modulus $|z-z_1|^2$ is simply the square of the distance between $z$ and $z_1$. Using the distance formula, this is $(x-2)^2 + (y-3)^2$. Similarly, $|z-z_2|^2$ becomes $(x-3)^2 + (y-4)^2$.

Now, let us calculate the right-hand side, $|z_1-z_2|^2$. Subtracting the coordinates, we get $(2-3) + i(3-4) = -1 - i$. The square of its modulus is $(-1)^2 + (-1)^2 = 2$.

So, our equation is:

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

### The Moment of Cancellation

Now, do not be intimidated by the expansion. It is here that the magic happens.

Expanding the first term, we get $x^2 - 4x + 4 + y^2 - 6y + 9$, which simplifies to $x^2 + y^2 - 4x - 6y + 13$. Expanding the second term, we get $x^2 - 6x + 9 + y^2 - 8y + 16$, which simplifies to $x^2 + y^2 - 6x - 8y + 25$.

When we subtract the second from the first, the quadratic terms $x^2$ and $y^2$ vanish entirely! This is the beauty of the problem.

We are left with the following linear expression:

$$(-4x - (-6x)) + (-6y - (-8y)) + (13 - 25) = 2$$

This simplifies to $2x + 2y - 12 = 2$.

### The Final Revelation

Our massive equation has collapsed into the simple linear form $2x + 2y = 14$, or $x + y = 7$. This is the equation of a straight line!

To find the sum of its intercepts, we rewrite it in the intercept form:

$$\frac{x}{7} + \frac{y}{7} = 1$$

The $x$-intercept is $7$, and the $y$-intercept is $7$. Adding them together, we get **14**.

You have successfully navigated the complex plane and arrived at the solution. Remember, every complex problem is just a geometric story waiting to be told.

Animated Solution for Mathematics - 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

Select Answer:

Visualized Solution

Visualizing $z_{1}$ and $z_{2}$

Defining $z$ and the Distance Formula

Calculating $∣ z_{1} - z_{2} ∣^{2}$

Setting up the Locus Equation

Expanding $∣ z - z_{1} ∣^{2}$

Expanding $∣ z - z_{2} ∣^{2}$

Subtracting the Expressions

Simplifying to a Straight Line

Finding Sum of Intercepts

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Setup

The Algebraic Grind

The Moment of Cancellation

The Final Revelation

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List-I

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Animated Solution for Mathematics - Complex Numbers: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​=2+3i and z2​=3+4i. The set S={z∈C:∣z−z1​∣2−∣z−z2​∣2=∣z1​−z2​∣2} represents a