The equation ^2 = x^2 + y^2 2xy ,x,y, 0 is possible if - Vidyadip

MCQ

The equation ${\sin ^2}\theta = \frac{{{x^2} + {y^2}}}{{2xy}},x,y, \ne 0$ is possible if

✓
$x = y$
B
$x = \, -y$
C
$2x = y$
D
none of these

✓

Answer

Correct option: A.

$x = y$

a
Now, $\sin ^{2} \theta=\frac{x^{2}+y^{2}}{2 x y}$

$\therefore \mathrm{x},$ $y$ have same sign

$\frac{x^{2}+y^{2}}{2 x y}=\frac{1}{2}\left[(\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}})^{2}+2\right] \geq 1$

Now, $\sin ^{2} \theta \leq 1 .$

Therefore, $\frac{x^{2}+y^{2}}{2 x y}=1$

$ \Rightarrow x=y$

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