Question

Express the relation of $a$ and $b$ if $\sin \theta =\frac{2ab}{a^2+b^2}$ where $a>0,b>0$

• A. $a>0,b>0$
• B. $a<0,b>0$
• C. $a>0,b<0$
• D. $a<0,b<0$

Answer 1

Answer: (A)

$a>0,b>0$

We know that ${\sin }^2\theta \le 1$
i.e. ${\left(\frac{2ab}{a^2+b^2}\right)}^2\le 1$
or $4a^2{b}^2\le {(a^2+b^2)}^2$
or $4a^2{b}^2-{(a^2+b^2)}^2\le 0$
or ${(a^2+b^2)}^2-4a^2{b}^2\ge 0$
or $(a^2-b^2)\ge 0$
or $a^2\ge b^2$
or $a\ge b$
$\therefore a>0,b>0$