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 \(2kx + hy = 3hk.\)