Question

The value of \(\rm \sqrt{(5 + 12i) }\), Where i = \(\sqrt {-1}\)
1. ± (2 + 3i)
2. ± (3 + 2i)
3. ± (2 - 3i)
4. ± (1 + 2i)

Answer 1

Correct Answer - Option 2 : ± (3 + 2i)

Concept:

(x + y)² = (x - y)² + 4xy

 

Calculation:

Let, \(\rm \sqrt{(5 + 12i) }\) = x + iy

On squaring both sides we get,

5 + 12i = (x + iy)² = (x² - y²) + i (2xy)

On comparing real and imaginary parts on both sides we get

(x2 - y2) = 5 and 2xy = 12⇒ xy = 6

\(\rm (x^2+y^2)=\sqrt{(x^2-y^2)+4x^2y^2}=\sqrt{(5)^2+4\times 36}=\sqrt{169}=13\)           (∵ (x + y)2 = (x - y)2 + 4xy)

Now, (x2 - y2) = 5 and (x2 + y2) = 13

⇒ 2x² = 18 

⇒ x² = 9 

⇒ x = ±3

And, 9 - y² = 5 

⇒ y2 = 4

 ⇒ y = ± 2

Hence, \(\rm \sqrt{(5 + 12i) }\) = ± (3 + 2i)

Hence , option (2) is correct.