Question:
Let the equation $x^{2}+y^{2}+p x+(1-p) y+5=0$ represent circles of varying radius $r \in(0,5]$. Then the number of elements in the set $S=\left\{q: q=p^{2}\right.$ and $\mathrm{q}$ is an integer $\}$ is
Solution:
$r=\sqrt{\frac{p^{2}}{4}+\frac{(1-p)^{2}}{4}-5}=\frac{\sqrt{2 p^{2}-2 p-19}}{2}$
Since, $r \in(0,5]$
So, $0<2 p^{2}-2 p-19 \leq 100$
$\Rightarrow \mathrm{p} \in\left[\frac{1-\sqrt{239}}{2}, \frac{1-\sqrt{39}}{2}\right) \cup\left(\frac{1+\sqrt{39}}{2}, \frac{1+\sqrt{239}}{2}\right]$ so, number of integral values of $\mathrm{p}^{2}$ is 61