Question

If z = x +iy then the equation |z + 1| = |z - 1| represents

• A. a circle
• B. a parabola
• C. x-axis
• D. y-axis

Answer 1

The Correct Option is D

If \(z = x + iy\), then the equation \(|z + 1| = |z - 1|\) represents

Let \(z = x + iy\). The given equation is \(|z + 1| = |z - 1|\).

Substituting \(z = x + iy\), we get \(|x + iy + 1| = |x + iy - 1|\).

\(|(x+1) + iy| = |(x-1) + iy|\)

Taking the magnitude of both sides gives: \(\sqrt{(x+1)^2 + y^2} = \sqrt{(x-1)^2 + y^2}\)

Squaring both sides: \((x+1)^2 + y^2 = (x-1)^2 + y^2\)

\(x^2 + 2x + 1 + y^2 = x^2 - 2x + 1 + y^2\)

\(2x = -2x\)

\(4x = 0\)

\(x = 0\)

This represents the y-axis.

Answer: (D) y-axis

Answer 2

We have:

$$ |z + 1| = |z - 1| \implies |x + iy + 1| = |x + iy - 1|. $$

Simplify each modulus:

$$ |(x+1) + iy| = |(x-1) + iy| \implies \sqrt{(x+1)^2 + y^2} = \sqrt{(x-1)^2 + y^2}. $$

Square both sides:

$$ (x+1)^2 + y^2 = (x-1)^2 + y^2. $$

Expand and simplify:

$$ x^2 + 2x + 1 + y^2 = x^2 - 2x + 1 + y^2 \implies 4x = 0 \implies x = 0. $$

This represents the y-axis.

Final Answer: The final answer is $ {\text{y-axis}} $.