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 $a\in \left(-4,3\right)$
$\therefore a\in (-1,3)$