Question

Find the equation of the line which passes through the point $(3,4)$ and is such that the portion of it intercepted between the axes is divided by the point in the ratio $2:3$.

Answer 1

Since, $\text{The equation of the line with intercepts}\text{a and b is}\frac{x}{a}+\frac{y}{b}=1$......(i)
$\text{This line intersects the axes at A (a, 0) and B (0, b)}$.
The point $P(3,4)$ divides the line segment $AB$ in the ratio of $2:3$ as show in the figure.

So, $P(3,4)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$ [From section formula]
Where, $m=2,n=3,(x_1,y_1)=(a,0)$ and $(x_2,y_2)=(0,b)$
Thus, $P(3,4)=(\frac{(2\times 0)+(3\times a)}{2+3},\frac{(2\times b)+(3\times 0)}{2+3})$
$\Rightarrow (3,4)=(\frac{3a}{5},\frac{2b}{5})$
$\Rightarrow 3=\frac{3a}{5}$ and $4=\frac{2b}{5}$
$\Rightarrow 15=3a$ and $20=2b$
$\Rightarrow a=\frac{15}{3}$ and $b=\frac{20}{2}$
$\therefore a=5,b=10$
Hence, the equation of the line is $\frac{x}{5}+\frac{y}{10}=1$.
[Substitute $a=5,b=10$ in equation(i)]