Question

The value of $p$ for which the equation $x^2+pxy+y^2-5x-7y+6=0$ represents a pair of straight lines, is

• A. $\frac{5}{2}$
• B. $5$
• C. $2$
• D. $\frac{2}{5}$

Answer 1

The correct option is A

$\frac{5}{2}$

Explanation for the correct answer:

Compute the required value:

The given equation is $x^2+pxy+y^2-5x-7y+6=0$ $...\left(i\right)$

The general form of a second-degree equation is given as

$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ $...\left(ii\right)$

Comparing equations $\left(i\right)$ and $\left(ii\right)$ we get

$a=1,h=\frac{p}{2},b=1,g=\frac{-5}{2},f=\frac{-7}{2},c=6$

The condition for a general second-degree equation to be a pair of straight lines is

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

Substituting all the required values we get,

$6+2\times \frac{-7}{2}\times \frac{-5}{2}\times \frac{p}{2}-{\left(\frac{-7}{2}\right)}^2-{\left(\frac{-5}{2}\right)}^2-6{\left(\frac{p}{2}\right)}^2=0$

$\Rightarrow$ $6+\frac{35p}{4}-\frac{49}{4}-\frac{25}{4}-6\frac{p^2}{4}=0$

$\Rightarrow$ $24+35p-74-6p^2=0$

$\Rightarrow$ $6p^2+35p-50=0$

$\Rightarrow$ $(2p-5)(3p-10)=0$

$\Rightarrow$ $p=\frac{5}{2}$ or $p=\frac{10}{3}$

Hence from the given options the value of $p$ is $\frac{5}{2}$.

Hence, option $\left(A\right)$ is the correct answer.