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

Answer:

x² + 2y² = 2

Step-by-step 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

$(\frac{x}{\sqrt{2} } )^{2} +y^{2} = \cos^{2}\phi + \sin^{2}\phi =1$

⇒ x² + 2y² = 2.

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