 # A straight line passes through the point Question:

A straight line passes through the point (5, -2) and the portion of the line intercepted between the axes is divided at this point in the ratio 2 : 3. Find the equation of the line.

Solution:

Given : The ratio of the line intercepted between the axes is 2 :3

Let $\left(x_{1}, y_{1}\right)=A(a, 0)$

And $\left(x_{2}, y_{2}\right)=B(0, b)$

Where a and b are intercepts of the line.

Formula used:

The equation of the line is : $\frac{x}{a}+\frac{y}{b}=1$

And the co-ordinate axis is divided at (5,-2) , thus by using Section formula

$(x, y)=\left(\frac{m y_{1}+n x_{1}}{m+n}, \frac{m y_{2}+n x_{2}}{m+n}\right)$

$=\left(\frac{2 * 0+3 \mathrm{a}}{5}, \frac{2 \mathrm{~b}+3 * 0}{5}\right)=\left(\frac{3 \mathrm{a}}{5}, \frac{2 \mathrm{~b}}{5}\right)$

(5,-2) divides the co-ordinate axis, thus (x,y)= (5,-2).

$\frac{3 a}{5}=5 \Rightarrow a=25 / 3, \frac{2 b}{4}=-2 \Rightarrow b=-5$

Equation of the line becomes $\frac{x}{a}+\frac{y}{b}=1$

$\frac{x}{25 / 3}+\frac{y}{-5}=1$

$\frac{3 x}{25}-\frac{y}{5}=1$

$\frac{3 x-5 y}{25}=1$

Hence ,3x – 5y = 25 is the required equation of the line.  