Question

The equation \(z^2+\bar{z}^2=4\) in the complex plane (where \(\bar{z}\) is the complex conjugate of z) represents

• A. Ellipse
• B. Hyperbola
• C. Circle of radius 2
• D. Circle of radius 4

Answer 1

The Correct Option is B

To solve the problem, we need to interpret the given equation in the context of the complex plane. The given equation is:

\(z^2 + \bar{z}^2 = 4\)

Let \(z = x + yi\) where \((x, y)\) are real numbers and \(\bar{z}\) is the complex conjugate, i.e., \(\bar{z} = x - yi\).

Substituting these into the equation:

\(z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi\)

\(\bar{z}^2 = (x - yi)^2 = x^2 - y^2 - 2xyi\)

Adding these:

\(z^2 + \bar{z}^2 = (x^2 - y^2 + 2xyi) + (x^2 - y^2 - 2xyi) = 2(x^2 - y^2)\)

Thus, the equation simplifies to:

\(2(x^2 - y^2) = 4\)

Dividing through by 2 gives:

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

This is the standard form of a hyperbola in the complex plane. Hence, the correct interpretation of the equation \(z^2 + \bar{z}^2 = 4\) is a hyperbola.

Correct Answer: Hyperbola