The number of integral values of \(p\) in the domain \([-5,5]\), such that the equation \(2 x^2+4 x y-\mathrm{p} y^2+4 x+\mathrm{q} y+1=0\) represents pair of lines, are

Given equation of pair of lines is
\(2 x^2+4 x y-p y^2+4 x+q y+1=0\)
Comparing with \(\mathrm{a} x^2+2 \mathrm{~h} x y+\mathrm{b} y^2+2 \mathrm{~g} x+2 \mathrm{f} y+\mathrm{c}=0\), we get \(\mathrm{a}=2, \mathrm{~h}=2, \mathrm{~b}=-\mathrm{p}\)
If the given equation represents a pair of straight lines, then
\(\begin{aligned}
& \mathrm{h}^2 \geq \mathrm{ab} \\
& \Rightarrow 4 \geq-2 \mathrm{p} \\
& \Rightarrow 2 \geq-\mathrm{p} \\
& \Rightarrow \mathrm{p} \geq-2
\end{aligned}\)
\(\therefore \quad\) Possible values of \(\mathrm{p}\) from domain \([-5,5]\) are \(-2,-1,0,1,2,3,4,5\).
\(\therefore \quad\) Number of integral values of \(p=8\)