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 $ 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{6{{p}^{2}}}{4}=0 $ $ \Rightarrow $ $ 24+35p-74-6{{p}^{2}}=0 $ $ \Rightarrow $ $ 6{{p}^{2}}-20p-15p+50=0 $ $ \Rightarrow $ $ (2p-5)(3p-10)=0 $ $ \Rightarrow $ $ p=\frac{5}{2},\frac{10}{3} $