Question

A region in the $xy$-plane is bounded by the curve $y = \sqrt{25-x^2} $ and the line $ y = 0 $ .If the point $ (a, a +1) $ lies in the interior of the region, then

• A. $ a \in \left(-4,3\right) $
• B. $ a \in (-\infty , -1)\cup(3,\infty) $
• C. $ a \in (-1 , 3) $
• D. None of these

Answer 1

Answer: (C)

If $(a, a+1)$ lies in the region $y=\sqrt{25-x^{2}}$ and line $y=0$
Then, $a+1 >\,0$ and $a+1 <\, \sqrt{25-a^{2}}$

$\Rightarrow a \,> -1$ and $(a+1)^{2}<\, 25-a^{2}$
$\Rightarrow a \,> -1$ and $2a^{2}+2a-24<\,0$
$\Rightarrow a>\, -1$ and $a^{2} + a-12<\,0$
$\Rightarrow a>\, -1$ and $(a-3)(a+4)<\,0$
$\Rightarrow a>\, -1$ and $a \in\left(-4,3\right)$
$\therefore a \in (-1, 3)$