Question

Find the equation to the straight line which passes through the point (−4, 3) and is such that the portion of it between the axes is divided by the point in the ratio 5 : 3.

Answer 1

The equation of the line with intercepts a and b is $\frac{x}{a}+\frac{y}{b}=1$

The line $\frac{x}{a}+\frac{y}{b}=1$ intersects the axes at (a, 0) and (0, b).

So, (−4, 3) divides the line segment AB in the ratio 5 : 3

$$\begin{gathered} \therefore -4=\frac{5\times 0+3\times a}{5+3},3=\frac{5\times b+3\times 0}{5+3} \\ \Rightarrow 3a=-32,5b=24 \\ \Rightarrow a=-\frac{32}{3},b=\frac{24}{5} \end{gathered}$$

Thus, the equation of the required line is

$$\begin{gathered} \frac{x}{-{\displaystyle \frac{32}{3}}}+\frac{y}{{\displaystyle \frac{24}{5}}}=1 \\ \Rightarrow \frac{3x}{-{\displaystyle 32}}+\frac{5y}{{\displaystyle 24}}=1 \\ \Rightarrow 9x-20y+96=0 \end{gathered}$$