Question

If \( z = x + iy \) and the point \( P \) represents \( z \) in the Argand plane, then the locus of \( z \) satisfying the equation \( |z-1| + |z+i| = 2 \) is:

• A. \( 15x^2 - 2xy + 15y^2 - 16x + 16y - 48 = 0 \)
• B. \( 3x^2 + 2xy + 3y^2 - 4x - 4y = 0 \)
• C. \( 3x^2 - 2xy + 3y^2 - 4x + 4y = 0 \)
• D. \( 15x^2 + 2xy + 15y^2 + 16x - 16y - 48 = 0 \) \bigskip

Hint:

When solving geometric locus problems, apply the distance formula and simplify step by step to derive the equation of the locus.

Answer 1

The Correct Option is C

Step 1: The equation is \( |z - 1| + |z + i| = 2 \). This represents the sum of the distances from the point \( z = x + iy \) to the points \( 1 \) and \( -i \) in the complex plane. This equation defines an ellipse with foci at \( 1 \) and \( -i \), and the length of the major axis is \( 2 \). 

Step 2: We can express the distances as: \[ |z - 1| = \sqrt{(x - 1)^2 + y^2}, \quad |z + i| = \sqrt{x^2 + (y + 1)^2}. \] Thus, the equation becomes: \[ \sqrt{(x - 1)^2 + y^2} + \sqrt{x^2 + (y + 1)^2} = 2. \] 

Step 3: By simplifying the equation and squaring both sides, we obtain the equation of the ellipse: \[ 3x^2 - 2xy + 3y^2 - 4x + 4y = 0. \]