Question

Find the equations of the straight lines which pass through the origin and trisect the portion of the straight line $2x+3y=6$ which is intercepted between the axes.

Answer 1

Given line

$2x+3y=6$

$\therefore x$ intercept $(3,0)$

$y$ intercept $(0,2)$

required lines are $l_1,l_2$

Let pt $A,B$ trisect the line $PQ$

$\overline{PQ}=\sqrt{4+9}=\sqrt{13}$

Note that eqn. of $l_1$ of the form $y=m_{1}x$

Note that eqn. of $l_2$ of the form $y=m_{2}x$

where $m_1,m_2$ are the slops.

Using section formula vote $PB=2BQ$ or $PB:BQ=2:1$

So coordinate of $B$

$\begin{array}{c}x:\frac{2\times 3+1\times 0}{3}=2\\ y:\frac{2\times 0+1\times 2}{3}=\frac{2}{3}\end{array}\}B=(2,\frac{2}{3})$

Similarly $A$ which is mid point of $PB$

$\begin{array}{c}x:\frac{0+2}{2}=1\\ y:\frac{2+\frac{2}{3}}{2}=\frac{4}{3}\end{array}\}A=(1,\frac{1}{3})$

$l_1:y=m_{1}x$ t point $A\frac{4}{3}=m_{1}1$ or $m_1=\frac{4}{3}$

$l_2:y=m_{2}x$ at point $B\frac{2}{3}=m_{2}2$ or $m_2=\frac{1}{3}$

or $l_1:y=\frac{4}{3}x$ and $l_2:y=\frac{1}{3}x$