Question

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

• A. 5/2
• B. 5
• C. 2
• D. 2/5

Answer 1

Answer: (A)

Key Idea: The equation $a{x}^2+b{y}^2+2hxy+2gx+2fy+c=0$ will represent a pair of straight lines, if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$ . Given equation $x^2+pxy+y^2-5x-7y+6=0$ will represent a pair of straight lines, if $\mathrm{1.1.6}+2\left(-\frac{7}{2}\right)\left(\frac{-5}{2}\right)\left(\frac{p}{2}\right)-1{\left(\frac{-7}{2}\right)}^2$ $-1{\left(\frac{-5}{2}\right)}^2-6{\left(\frac{p}{2}\right)}^2=0$ $\Rightarrow$ $6+\frac{35p}{4}-\frac{49}{4}-\frac{25}{4}-\frac{6p^2}{4}=0$ $\Rightarrow$ $24+35p-74-6p^2=0$ $\Rightarrow$ $6p^2-20p-15p+50=0$ $\Rightarrow$ $(2p-5)(3p-10)=0$ $\Rightarrow$ $p=\frac{5}{2},\frac{10}{3}$