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,
$(h,k)=(\frac{1\times 0+2\times x}{1+2},\frac{1\times y+2\times0}{1+2})$

$⇒(h,k)=(\frac{2x}{3},\frac{y}{3})$

$⇒h=\frac{2x}{3} \space and k=\frac{y}{3}$

$⇒x=\frac{3h}{2} and\space y=3k$
Therefore, the respective coordinates of A and B are $(\frac{3h}{2},0)$ and $(0, 3k)$.
Now, the equation of line AB passing through points $(\frac{3h}{2},0)$and $(0, 3k)$ is
$(y-0)=\frac{3k-0}{0-\frac{3h}{2}}(x-\frac{3h}{2})$
$y=-\frac{2k}{h}(x-\frac{3h}{2})$
$hy=-2kx+3hk$
$i.e,2kx+hy=3hk$
Thus, the required equation of the line is $2kx + hy = 3hk.$