Question

$sin \,\theta = \frac{1}{2} \left(\sqrt{\frac{x}{y} } + \sqrt{\frac{y}{x}}\right)$ necessarily implies :

• A. x > y
• B. x < y
• C. x = y
• D. both x and y are purely imaginary

Answer 1

Answer: (C)

Put $t = \sqrt{\frac{x}{y}} > 0,$
Consider $t+\frac{1}{t} = \left(\sqrt{t} - \frac{1}{\sqrt{t}}\right)^{2} + 2 \ge 2$ equality holding
iff $t = 1$
Also, $t+\frac{1}{t} = 2 \,sin \,\theta \,\le \,2$, so that t should necessarily
be 1, i.e., $x = y.$