Find the equation of the line $PQ$ passing through the points $\left(5, 0\right)$ and $\left(0, -6\right)$. Also find the equation of another line $AB$ intersecting $PQ$ at right angles and passing through the point $\left(6,1\right)$.
Let $P \left(x_1, y_1\right) = \left(5,0\right)$; $\;$ $Q \left(x_2, y_2\right) = \left(0, -6\right)$
Slope of $PQ = m_1 = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-6 - 0}{0 - 5} = \dfrac{6}{5}$
Equation of $PQ$ is: $\;$ $y - y_1 = m_1 \left(x - x_1\right)$
i.e. $\;$ $y - 0 = \dfrac{6}{5} \left(x - 5\right)$
i.e. $5y = 6x - 30$
$\therefore \;$ Equation of $PQ$ is: $\;$ $6x - 5y = 30$
$\because \;$ $AB \perp PQ$ $\implies$ Slope of $AB = m_2 = - \dfrac{1}{m_1} = -\dfrac{5}{6}$
$AB$ passes through the point $\left(6, 1\right)$
$\therefore \;$ Equation of $AB$ is: $\;$ $y - 1 = - \dfrac{5}{6} \left(x - 6\right)$
i.e. $\;$ $6 y - 6 = - 5 x + 30$
$\therefore \;$ Equation of $AB$ is: $\;$ $5x + 6y = 36$