jmc

Question

The complex number z traces a circle centered at the origin with radius 2. Then z + 1z traces a: (A) circle (B) parabola (C) ellipse (D) hyperbola Enter the letter of the correct option.

Accepted Answer

Let z = a + bi, where a and b are real numbers. Since |z| = 2, a^2 + b^2 = 4. Then align* z + 1z &= a + bi + 1a + bi &= a + bi + 1a + bi &= a + bi + a - bia^2 + b^2 &= a + bi + a - bi4 &= 54 a + 34 bi. align*Let x + yi = z + 1z, so x = 54 a and y = 34 b. Then x^2(5/4)^2 + y^2(3/4)^2 = a^2 + b^2 = 4,so x^2(5/2)^2 + y^2(3/2)^2 = 1.Thus, z + 1z traces an ellipse. The answer is (C).