Question

Find an algebraic equation for all the points in the coordinate plane traced by the complex numbers

z = √2 cos(theta) + i sin(theta).

Answer 1

x² + 2y² = 2

Explanation:

The general form of a complex number z in the Cartesian coordinate plane is given by z = x + iy ........ (1)

Now, the given complex number is z = √2 CosФ + i SinФ ....... (2)

Hence, comparing equations (1) and (2), we get, x = √2 CosФ and y = SinФ

Now, we can eliminate Ф to combine the above two equations as

`((x)/(√(2) ) )^(2) +y^(2) = \cos^(2)\phi + \sin^(2)\phi =1`

⇒ x² + 2y² = 2.

Therefore, this is the algebraic equation required, which is the path traced by the given complex number z. (Answer)