Let the equation $x^2+y^2+px+\left(1-p\right)y+5=0$ represent circles of varying radius $r\in \left(0,5\right]$. Then the number of elements in the set $\mathrm{S}=\left\{\mathrm{q}:\mathrm{q}={\mathrm{p}}^2\mathrm{and}\mathrm{q}\mathrm{is}\mathrm{an}\mathrm{integer}\right\}$

$r=\sqrt{\frac{p^2}{4}+\frac{(1-p{)}^2}{4}-5}$

Given  $0<\mathrm{r}\le 5\Rightarrow 0<{\mathrm{r}}^2\le 25$

$\Rightarrow 0<\frac{{\mathrm{p}}^2}{4}+\frac{(1-p{)}^2}{4}-5\le 25\Rightarrow$$0<2p^2-2p-19\le 100$

$\text{(i) }2p^2-2p-19\le 100\Rightarrow$$1-\sqrt{239}\le \mathrm{p}\le 1+\sqrt{239}$ .........(1) 

$\text{(ii) }2p^2-2p-19>0\Rightarrow$ $\mathrm{p}<1-\sqrt{39}\text{or}\mathrm{p}>1+\sqrt{39}$ ............(2)

$\left(1\right)\cap \left(2\right)\Rightarrow \mathrm{p}\in [1-\sqrt{239},1-\sqrt{39})$$\cup (1+\sqrt{39},1+\sqrt{239}]$

$\therefore$ No.of integral values of  $q=p^2 \text{is }61$