Find the equation of the straight line which passes through the point $\left(-5, 4\right)$ and is such that the portion of it intercepted between the axes is divided by the point in the ratio $1 : 2$
Let
intercept made by the required line on the $X$ axis $= a$
intercept made by the required line on the $Y$ axis $= b$
Then, the required line cuts the $X$ axis at the point $A \left(a, 0\right)$ and the $Y$ axis at the point $B \left(0, b\right)$.
The required equation passes through the point $P\left(-5, 4\right)$.
Given: $\;$ The point $P$ divides the line join of points $A$ and $B$ internally in the ratio $1 : 2$.
$\therefore \;$ By section formula for internal division, we have,
for the $x$ coordinate: $\;\;$ $-5 = \dfrac{1 \times 0 + 2 \times a}{1 + 2}$ $\implies$ $- 15 = 2a$ $\implies$ $a = \dfrac{-15}{2}$
and for the $y$ coordinate: $\;\;$ $4 = \dfrac{1 \times b + 2 \times 0}{1 + 2}$ $\implies$ $b = 12$
The equation of the required line be of the form: $\;$ $\dfrac{x}{a} + \dfrac{y}{b} = 1$
$\dfrac{x}{-15 / 2} + \dfrac{y}{12} = 1$
i.e. $\;$ $\dfrac{2x}{-15} + \dfrac{y}{12} = 1$
i.e. $\;$ $24x - 15y = - 180$
$\therefore \;$ The equation of the required line is
$8x - 5y + 60 = 0$