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. $aϵ[-4,3]$
• B. $aϵ(-\infty ,-1]\cup [3,+\infty )$
• C. $aϵ(-1,3)$
• D. $aϵ[-1,3]$

Answer 1

Answer: (C)

$aϵ(-1,3)$

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