Explanation

(b)
The given equation
$x^{2} + kxy + y^{2} - 5x - 7y + 6 = 0$ is compared with
${ax}^{2} + 2hxy + {by}^{2} + 2gx + 2fy + c = 0$, we get
$a = 1,\ b = 1,\ h = \frac{k}{2},\ g = \frac{- 5}{2},\ f = \frac{- 7}{2},\ c = 6$
This equation represents a pair of straight lines,
$if\ \left| \begin{matrix}
a & h & g \\
h & b & f \\
g & f & c
\end{matrix} \right| = 0$
$\Rightarrow \left| \begin{matrix}
1 & k/2 & - 5/2 \\
k/2 & 1 & - 7/2 \\
- 5/2 & - 7/2 & 6
\end{matrix} \right| = 0$
$\Rightarrow 1\left( 6 - \frac{49}{4} \right) - \frac{k}{2}\left( \frac{6k}{2} - \frac{35}{4} \right) - \frac{5}{2}\left( - \frac{7k}{4} + \frac{5}{2} \right) = 0$
$\Rightarrow \left( \frac{24 - 49}{4} \right) - \frac{k}{2}\left( \frac{12k - 35}{4} \right) - \frac{5}{2}\left( \frac{- 7k + 10}{4} \right) = 0$
$\Rightarrow \ - 50 - 12k^{2} + 35k + 35k - 50 = 0$
$\Rightarrow \ {- 12k}^{2} + 70k - 100 = 0$
$\Rightarrow \ {6k}^{2} - 35k + 50 = 0$
$\Rightarrow k = \frac{10}{3}$