Explanation

(a)
Given equation
$x^{2} + pxy + y^{2} - 5x - 7y + 6 = 0$
Will represent a pair of straight lines, if
$1 \bullet 1 \bullet 6 + 2\left( - \frac{7}{2} \right)\left( \frac{- 5}{2} \right)\left( \frac{p}{2} \right) - 1\left( \frac{- 7}{2} \right)^{2} - 1\left( \frac{- 5}{2} \right)^{2} - 6\left( \frac{p}{2} \right)^{2} = 0$
$\Rightarrow \ 6 + \frac{35p}{4} - \frac{49}{4} - \frac{25}{4} - \frac{{6p}^{2}}{4} = 0$
$\Rightarrow \ 35p - 50 - {6p}^{2} = 0$
$\Rightarrow \ (2p - 5)(3p - 10) = 0$
$\Rightarrow \ p = \frac{5}{2},\frac{10}{3}$