Question

For what value of $‘p^{\prime },y^2+xy+p{x}^2-x-2y+p=0$ represent $2$ straight lines :

• A. $2$
• B. $\frac{2}{3}$
• C. $\frac{1}{4}$
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

We have the equation $y^2+cy+p{x}^2-x-2y+p=0$
We know any general equation
$a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0...\left(1\right)$
represents two straight lines if
$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0...\left(2\right)$
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 $\left(2\right)$
$p\times 1\times p+2\times -1\times -\frac{1}{2}\times \frac{1}{2}-p\times {\left(-1\right)}^2-1\times {\left(-\frac{1}{2}\right)}^2-p\times {\left(\frac{1}{2}\right)}^2=0$
$\Rightarrow p^2+\frac{1}{2}-p-\frac{1}{4}-\frac{p}{4}=0\Rightarrow p^2-\frac{5p}{4}+\frac{1}{4}=0$
$\Rightarrow 4p^2-5p+1=0\Rightarrow \left(4p-1\right)\left(p-1\right)=0$
$\Rightarrow p=1,\frac{1}{4}$