Key Concepts and Formulas

• Distance in the Complex Plane: For complex numbers $z = x+iy$ and $z_0 = x_0 + iy_0$, the distance between them is $|z - z_0| = \sqrt{(x-x_0)^2 + (y-y_0)^2}$. Thus, $|z-z_0|^2 = (x-x_0)^2 + (y-y_0)^2$.
• Equation of a Straight Line: A linear equation of the form $ax + by = c$ represents a straight line in the Cartesian plane.
• Intercepts of a Straight Line: The x-intercept is the point where the line crosses the x-axis (y=0), and the y-intercept is the point where the line crosses the y-axis (x=0).

Step-by-Step Solution

Step 1: Represent Complex Numbers in Cartesian Coordinates

We represent the complex numbers as points in the Cartesian plane. Let $z = x + iy$, where $x$ and $y$ are real numbers. We are given $z_1 = 2 + 3i$ and $z_2 = 3 + 4i$. These correspond to the points $(2, 3)$ and $(3, 4)$ respectively.

Step 2: Calculate the Squared Moduli (Squared Distances)

We calculate the squared moduli required for the given equation.

• Calculating $|z - z_1|^2$: $z - z_1 = (x + iy) - (2 + 3i) = (x - 2) + i(y - 3)$ Therefore, $|z - z_1|^2 = (x - 2)^2 + (y - 3)^2$.

• Calculating $|z - z_2|^2$: $z - z_2 = (x + iy) - (3 + 4i) = (x - 3) + i(y - 4)$ Therefore, $|z - z_2|^2 = (x - 3)^2 + (y - 4)^2$.

• Calculating $|z_1 - z_2|^2$: $z_1 - z_2 = (2 + 3i) - (3 + 4i) = -1 - i$ Therefore, $|z_1 - z_2|^2 = (-1)^2 + (-1)^2 = 1 + 1 = 2$.

Step 3: Substitute and Simplify the Equation

Substitute the expressions for the squared moduli into the given equation: $|z - z_1|^2 - |z - z_2|^2 = |z_1 - z_2|^2$ $(x - 2)^2 + (y - 3)^2 - [(x - 3)^2 + (y - 4)^2] = 2$ Expand the squared terms: $(x^2 - 4x + 4) + (y^2 - 6y + 9) - (x^2 - 6x + 9) - (y^2 - 8y + 16) = 2$ Distribute the negative sign: $x^2 - 4x + 4 + y^2 - 6y + 9 - x^2 + 6x - 9 - y^2 + 8y - 16 = 2$ Combine like terms: $(x^2 - x^2) + (y^2 - y^2) + (-4x + 6x) + (-6y + 8y) + (4 + 9 - 9 - 16) = 2$ $0 + 0 + 2x + 2y - 12 = 2$ $2x + 2y - 12 = 2$

Step 4: Determine the Locus

Simplify the equation to find the locus: $2x + 2y = 2 + 12$ $2x + 2y = 14$ Divide by 2: $x + y = 7$ This equation represents a straight line.

Step 5: Calculate the Intercepts and their Sum

• x-intercept: Set $y = 0$: $x + 0 = 7 \implies x = 7$
• y-intercept: Set $x = 0$: $0 + y = 7 \implies y = 7$ The sum of the intercepts is $7 + 7 = 14$.

Interpretation and Evaluation of Options

The equation $x+y=7$ represents a straight line. The sum of its intercepts is 14. This corresponds to option (D).

Common Mistakes & Tips

• Sign Errors: Be extremely careful when distributing the negative sign in Step 3. This is a common source of errors.
• Expanding Squares: Ensure you expand the squared terms correctly using the formula $(a-b)^2 = a^2 - 2ab + b^2$.
• Simplify Carefully: Double-check each step of simplification to avoid mistakes in combining like terms.

Summary

The given equation simplifies to $x+y=7$, which represents a straight line. The x and y intercepts are both 7, so their sum is 14. Therefore, the set $S$ represents a straight line with the sum of its intercepts on the coordinate axes equal to 14.

Final Answer The final answer is \boxed{14}, which corresponds to option (D).