Question

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

Answer 1

Let AB be the line segment between the axes such that point R (h, k) divides AB in the ratio 1: 2.

Let the respective coordinates of A and B be (x, 0) and (0, y).

Since point R (h, k) divides AB in the ratio 1: 2, according to the section formula,

(h, k) = ${\displaystyle \frac{1\times 0+2\times x}{1+2},\frac{1\times y+2\times 0}{1+2}}$

= (h, k) = ${\displaystyle (\frac{2x}{3},\frac{y}{3})}$

= ${\displaystyle h=\frac{2x}{3}\text{and}k=\frac{y}{3}}$

= x = ${\displaystyle \frac{3h}{2}\text{and}y=3k}$

Therefore, the respective coordinates of A and B are ${\displaystyle (\frac{3h}{2},0)}$ and (0, 3k).

Now, the equation of line AB passing through points ${\displaystyle (\frac{3h}{2},0)}$ and (0, 3k) is

(y - 0) = ${\displaystyle \frac{3k-0}{0-\frac{3h}{2}}(x-\frac{3h}{2})}$

y = ${\displaystyle \frac{2k}{h}(x-\frac{3h}{2})}$

hy = ${\displaystyle -\frac{2k}{h}(x-\frac{3h}{2})}$

hy = -2kx + 3hk

i.e., 2kx + hy = 3hk

Thus, the required equation of the line is 2kx + hy = 3hk.