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Q.
For what value of $‘p' , y^2 + xy + px^2 - x - 2y + p = 0$ represent $2$ straight lines :
Straight Lines
Solution:
We have the equation $y^{2}+cy+px^{2}-x-2y+p=0$
We know any general equation
$ax^{2}+by^{2}+2hxy+2gx+2fy+c=0\,...\left(1\right)$
represents two straight lines if
$abc+2fgh-af^{2}-bg^{2}-ch^{2}=0\,...\left(2\right)$
On comparing given equation with $\left(1\right)$, We get
$a=p, b=1, h=\frac{1}{2}, g=-\frac{1}{2}, f=-1, c=p$
Put these value in equation $\left(2\right)$
$p\times1\times p+2\times-1\times-\frac{1}{2}\times\frac{1}{2}-p\times\left(-1\right)^{2}-1\times\left(-\frac{1}{2}\right)^{2}-p\times\left(\frac{1}{2}\right)^{2}=0$
$\Rightarrow p^{2}+\frac{1}{2}-p-\frac{1}{4}-\frac{p}{4}=0 \Rightarrow p^{2}-\frac{5p}{4}+\frac{1}{4}=0$
$\Rightarrow 4p^{2}-5p+1=0 \Rightarrow \left(4p-1\right)\left(p-1\right)=0$
$\Rightarrow p=1, \frac{1}{4}$