Question

If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.

Options

• 2

• `sqrt(3)`

• `sqrt(5)`

• 1

Answer 1

If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to `underlinebb(sqrt(5))`.

Explanation:

Given equation is `z + sqrt(2)|z + 1| + i` = 0

Put z = x + iy in the given equation.

`(x + iy) + sqrt(2)|x + iy + 1| + i` = 0

⇒ `x + iy + sqrt(2)[sqrt((x + 1)^2 + y^2)] + i` = 0

Now, equating real and imaginary part, we get

`x + sqrt(2) sqrt((x + 1)^2 + y^2` = 0 and y + 1 = 0

⇒ y = –1

⇒ `x + sqrt(2)sqrt((x + 1)^2 + (-1)^2` = 0  ...(∵ y = –1)

⇒ `sqrt(2)sqrt((x + 1)^2 + 1)` = –x

⇒ 2[(x + 1)² + 1] = x²

⇒ x² + 4x + 4 = 0

⇒ x = –2

Thus, z = –2 + i(–1) 

⇒ |z| = `sqrt(5)`