Question

If $p$ and $q$ are positive real numbers such that $p^{2}+q^{2}=1$, If the maximum value of $(p + q)$ is equal to $\sqrt{a}$ then a is equal to

Answer 1

Given that $p^{2}+q^{2}=1$
$\therefore p=cos\,\theta$ and $q = sin\,\theta$
Then $p+q=cos\,\theta+sin\,\theta$
We know that
$-\sqrt{a^{2}+b^{2}} \le\,a\,cos\,\theta+b\,sin\,\theta\,\le \sqrt{a^{2}+b^{2}}$
$\therefore -\sqrt{2}\le\,cos\,\theta\,\le\,\sqrt{2}$
Hence max. value of $p + q$ is $\sqrt{2}$