Question

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

• A. $p\in (–1,3)$
• B. $p\in (–\infty ,–1)\cup (3,\infty )$
• C. $p\in (–4,3)$
• D. $p\in (–\infty ,–1)$

Answer 1

The correct option is A

$p\in (–1,3)$

Explanation for the correct option:

Step 1. Draw the graph of given curves:

From the figure, we can see

For interior points, $OA<5$

$\Rightarrow$$\sqrt{{\left(p-0\right)}^2+{\left(p+1\right)}^2}<5$

$\Rightarrow$ $p^2+p^2+2p+1<25$

$\Rightarrow$ $2p^2+2p-24<25$

$\Rightarrow$ $p^2+p-12<0$

$\Rightarrow$ $\left(p+4\right)\left(p-3\right)<0$

$\therefore$$-4<p<3$

Step 2. In $1st$ and $2nd$ quadrant

$p+1>0$

$\Rightarrow$$p>-1$

$\Rightarrow -1<p<3$

Thus, $p\in (-1,3)$

Hence, Option $\left(A\right)$ is Correct.