Question

If $x+iy=\sqrt{\frac{3+4i}{5+12i}}$ then $169(x^2+y^2)=$

• A. $5$
• B. $10$
• C. $65$
• D. $50$

Answer 1

The correct option is C $65$

We have $x+iy=\sqrt{\frac{3+4i}{5+12i}}$,

Its complex conjugate is

$x-iy=\sqrt{\frac{3-4i}{5-12i}}$

Now multiplying $x+iywithx-iy,$

$\therefore$ $x^2+y^2=\sqrt{\frac{(3+4i)\times (3-4i)}{(5+12i)\times (5-12i)}}$

$\therefore$ $x^2+y^2=\sqrt{\frac{3^2+4^2}{5^2+{12}^2}}$

$\Rightarrow x^2+y^2$$=\frac{5}{13}$

Thus $169\times \frac{5}{13}=65.$