Question

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

Answer 1

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

The line passes through R (h, k).
∴ \[\frac{h}{a} + \frac{k}{b} = 1\]           ... (1)

The line intersects the coordinate axes at A (a, 0) and B (0, b).
Here, AP : PB = 1 : 2

\[\therefore h = \frac{1 \times 0 + 2 \times a}{1 + 2}, k = \frac{1 \times b + 2 \times 0}{1 + 2}\]

\[ \Rightarrow a = \frac{3h}{2}, b = 3k\]

Substituting 

\[a = \frac{3h}{2}, b = 3k\] in \[\frac{x}{a} + \frac{y}{b} = 1\] 

\[\frac{2x}{3h} + \frac{y}{3k} = 1\]

\[ \Rightarrow 2kx + hy - 3hk = 0\]

Hence, the equation of the line is \[2kx + hy - 3hk = 0\]