Question

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

• A. $a\in (-5,5)$
• B. $a\in (-4,3)$
• C. $a\in (-1,5)$
• D. $a\in (-1,3)$

Answer 1

The correct option is D $a\in (-1,3)$

$y=\sqrt{25-x^2}$ from here $y\ge 0$
Now $y^2=25-x^2\Rightarrow x^2+y^2=25$
and $y=0$ bound a semi-circle above axis


$(a,a+1)$ lies in the interior of the region.
Clearly, $a+1>0\Rightarrow a>-1\dots \left(1\right)$
and $a^2+(a+1{)}^2-25<0$
$a^2+a-12<0\Rightarrow -4<a<3\dots \left(2\right)$
From $\left(1\right),\left(2\right)$ we get
$a\in (-1,3)$