Question

Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.

पर्याय

• `sqrt(10)`

• `7/2`

• `15/4`

• `2sqrt(3)`

Answer 1

Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is `underlinebb(7/2)`.

Explanation:

Let z = x + iy

Then, `|(z - i)/(z + 2i)|` = 1

⇒ x² + (y – 1)² = x² + (y + 2)²

⇒ –2y + 1 = 4y + 4

⇒ 6y = –3

⇒ y = `-1/2`

∵ |z| = `5/2`

⇒ x² + y² = `25/4`

⇒ x² = `24/4` = 6

∴ z = x + iy

⇒ z = `± sqrt(6) - i/2`

|z + 3i| = `sqrt(6 + 25/4)`

= `sqrt(49/4)`

⇒ |z + 3i| = `7/2`