Is the equation ${sec}^{2} θ = \frac{4 x y}{(x + y)^{2}}$ possible for real
value of $x$ and $y$ ?
If not, then find out a relation between $x$ and $y$ so that it may be possible.

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Solution

Since ${sec}^{2} θ \geq 1$ , we have from the given relation,
$\frac{4 x y}{(x + y)^{2}} \geq 1$ or $(x + y)^{2} \geq 4 x y$ .
or ${(x + y)}^{2} - 4 x y \leq 0$ or ${(x - y)}^{2} \leq 0$ .
Above is possible only when $x = y$ but $x \neq 0$ .

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