Point $P (h, k)$ divides a line segment between the axes in the ratio $1 : 2$ , find the equation of the line.

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Solution

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Let the $A B$ be a line between axis
& point $R (h, k)$ divides $A B$ in ratio $1 : 2$

Let $A B$ makes x- intercept a & y- intercept $b$

point $A (a, 0) & B (0, b)$

So the equation of line $A B$ by intercept form is

$\frac{x}{a} + \frac{y}{b} = 1$ ___(i)

Coordinates of point which divide a line segment

joining two points $(x_{1}, y_{1}) & (x_{2}, x_{2})$ in ratio
of $m_{1} : m_{2}$ are

$= [\frac{m_{2} x_{2} + m_{1} x_{1}}{m_{1} + m_{2}}, \frac{m_{2} y_{2} + m_{1} y_{1}}{m_{1} + m_{2}}]$

point $R (h, k)$ divide $A B$ joining two points $(a, 0) & (0, b)$ in
the ratio $1 : 2$ .

$(h, k) = (\frac{1 \times 0 + 2 \times a}{1 + 2}, \frac{1 \times b + 2 \times 10}{1 + 2})$

$(h, k) = (\frac{0 + 2 a}{3}, \frac{b}{3})$

$(h, k) = \frac{2 a}{3}, \frac{b}{3} \Rightarrow \frac{2 a}{3} = h \Rightarrow 3 a = 3 h \Rightarrow a = \frac{3 h}{2}$

$\frac{b}{3} = k \Rightarrow b = 3 k$ putting a & b in (i) $\frac{x}{a} + \frac{y}{b} = 1$

$\frac{x}{\frac{3 h}{2}} + \frac{y}{3 k} = 1 \Rightarrow \frac{2 x}{3 h} + \frac{y}{3 k} = 1 \Rightarrow \frac{1}{3} (\frac{2 x}{h} + \frac{y}{k}) = 1$

$\frac{2 x (k) + y (b)}{h k} = 1 \times 3 \Rightarrow 2 k x + y b = 3 h k .$

This is required equation

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