Question

If square root of 8 - 15i is $\pm$ (a - ib). The a + b is _____

• A. Rational Number
• B. Irrational Number
• C. Integer
• D. Complex number

Answer 1

The correct option is B

Irrational Number

Let Square root of $8-15i=(a-ib)$
$\sqrt{8-15i}=a-ib$
$8-15i=a^2-b^2-2abi\dots \dots$ (1)
Compare the real and imaginary part on both sides
$a^2-b^2=8\dots \dots \left(2\right)$
$2ab=15\dots \dots \left(3\right)$
We know,
$a^2+b^2=\sqrt{(a^2-b^2{)}^2+4a^2{b}^2}$
$a^2+b^2=\sqrt{(8{)}^2+225}=\sqrt{64+225}=17$
$a^2+b^2=17\dots \dots \left(4\right)$
Solve equation 2 and equation 4
$a^2-b^2=8\dots \dots \left(2\right)$
$\underset{–}{a^2+b^2=17}\dots \dots \left(4\right)$
$2a^2=25$
$a^2=\frac{25}{2},a=\pm \frac{5}{\sqrt{2}}$
Substituting the value of a in equation 4
$\frac{25}{2}+b^2=17$
$b^2=17-\frac{25}{2}=\frac{9}{2}$
$b=\pm \frac{3}{\sqrt{2}}$