Find the ratio in which the line joining $A \left(6,5\right)$ and $B \left(4,-3\right)$ is divided by the line $y = 2$. Also, find the coordinates of the point of intersection.
Let any point on the line $y = 2$ be $\;$ $P \left(p, 2\right)$
Let point $P$ divide the line joining $A \left(6, 5\right)$ and $B \left(4, -3\right)$ in the ratio $k : 1$
Then by section formula,
$p = \dfrac{4k + 6}{k + 1}$ $\;\;\; \cdots \; (1a)$
$2 = \dfrac{-3k + 5}{k + 1}$ $\;\;\; \cdots \; (1b)$
Solving equation $(1b)$ for $k$ gives,
$2k +2 = -3k + 5$
i.e. $\;$ $5k = 3$ $\implies$ $k = \dfrac{3}{5}$
i.e. $\;$ The line $y = 2$ divides the line joining the points $A$ and $B$ in the ratio $3:5$
Substituting the value of $k$ in equation $(1a)$ gives
$p = \dfrac{4 \times \dfrac{3}{5} + 6}{\dfrac{3}{5} + 1} = \dfrac{42}{8} = \dfrac{21}{4}$
$\therefore \;$ The point of intersection is $\left(\dfrac{21}{4}, 2\right)$