Question

The locus of the variable point \( z = x + iy \) whose amplitude is always equal to \( \theta \), is:

• A. \( x^2 + y^2 = \tan^2 \theta \)
• B. \( y = x \tan \theta \)
• C. \( \frac{x^2}{\sin^2 \theta} + \frac{y^2}{\cos^2 \theta} = 1 \)
• D. \( \frac{x^2}{\sin^2 \theta} + \frac{y^2}{\cos^2 \theta} = 1 \)

Hint:

To find the locus of a point in the complex plane, use the amplitude of the complex number and relate it to the given conditions.

Answer 1

The Correct Option is B

The amplitude of the complex number \( z = x + iy \) is given by \( |z| = \sqrt{x^2 + y^2} \), and the amplitude is always equal to \( \theta \). Therefore, we have: \[ |z| = \sqrt{x^2 + y^2} = \tan \theta \] Squaring both sides: \[ x^2 + y^2 = \tan^2 \theta \] Thus, the locus of the point is given by \( x^2 + y^2 = \tan^2 \theta \). Therefore, the correct answer is \( x^2 + y^2 = \tan^2 \theta \).