Question

If the intercept of a line between coordinate axes is divided by the point (-5, 4) in the ratio $1:2$, then find the equation of the line.

Answer 1

It is given that a point (-5,4) divides an intercept of a line in a ratio $1:2$. Let us assume that the x-intercept of the line will have coordinates (x, 0) and the y-intercept of the line will have the coordinates (0, y).
Apply section formula $\left( x,y \right)=\left( \dfrac{m\times {{x}_{2}}+n\times {{x}_{1}}}{m+n},\dfrac{m\times {{y}_{2}}+n\times {{y}_{1}}}{m+n} \right)$,(where $m:n$ is the given ratio and (x, y) is the point that divides the given line in the given ratio) to get values of x and y.
After getting the values of x and y, write the equation of line in intercept form, i.e.
$\dfrac{x}{a}+\dfrac{y}{b}=1$, where a and b are x-intercept and y-intercept respectively.

Complete step by step answer:
Let us assume that the x-intercept of the line will have coordinates (x, 0) and the y-intercept of the line will have the coordinates (0, y)
For the given ratio $1:2$ and the point (-5,4), we can use section formula $\left( x,y \right)=\left( \dfrac{m\times {{x}_{2}}+n\times {{x}_{1}}}{m+n},\dfrac{m\times {{y}_{2}}+n\times {{y}_{1}}}{m+n} \right)$ as:
Here we have ratio m:n as 1:2, (x,y) as (-5,4) and the coordinates as (x,0) and (0,y).

& \Rightarrow \left( -5,4 \right)=\left( \dfrac{1\times 0+2\times x}{1+2},\dfrac{1\times y+2\times 0}{1+2} \right) \\\ & \Rightarrow \left( -5,4 \right)=\left( \dfrac{2x}{3},\dfrac{y}{3} \right) \\\ \end{aligned}$$ By equating the coordinates of the point, we get: $\begin{aligned} & \Rightarrow -5=\dfrac{2x}{3}\text{ and }4=\dfrac{y}{3} \\\ & \Rightarrow x=-\dfrac{15}{2}\text{ and }y=12 \\\ \end{aligned}$ Now, to get the equation of line, we will use the intercept form, i.e. $\dfrac{x}{a}+\dfrac{y}{b}=1$ So, equation of line is: $\begin{aligned} & \Rightarrow \dfrac{x}{\left( -\dfrac{15}{2} \right)}+\dfrac{y}{12}=1 \\\ & \Rightarrow -\dfrac{2x}{15}+\dfrac{y}{12}=1 \\\ \end{aligned}$ Multiply both sides by 60, we get: $\begin{aligned} & \Rightarrow -8x+5y=60 \\\ & \Rightarrow 8x-5y+60=0 \\\ \end{aligned}$ **Hence $8x-5y+60=0$ is the required equation of line.** **Note:** Some might interchange the values of ratios given, i.e. m at place of n and vice versa. So, while applying the section formula, take care of writing the given ratio and co-ordinates at the right place or you might end up with a wrong answer. Also include the sign or the coordinates alongside as it might change the whole value.