Question

The intercept of a line between the coordinate axes is divided by the point $\left(-5,4\right)$ in the ratio $1:2$. The equation of the line will be

• A. $5x-8y+60=0$
• B. $8x-5y+60=0$
• C. $2x-5y+30=0$
• D. None of these

Answer 1

The correct option is B

$8x-5y+60=0$

Step 1: Find the values of $x-$intercept and $y-$intercept

Let us assume that, the $x-$intercept of the given straight line is $\left(a,0\right)$ and the $y-$intercept is $\left(0,b\right)$.

It is given that, the intercepts are divided by the point $\left(-5,4\right)$ in the ratio $1:2$.

The point, which divides the line segment in between $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ in the ratio $m:n$, can be given by $\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$.

Thus, $\left(-5,4\right)=\left[\frac{\left(1\times 0\right)+\left(2\times a\right)}{1+2},\frac{\left(1\times b\right)+\left(2\times 0\right)}{1+2}\right]$.

$\Rightarrow \left(-5,4\right)=\left(\frac{2a}{3},\frac{b}{3}\right)$.

So, $\frac{2a}{3}=-5$ and $\frac{b}{3}=4$.

Or, $a=-\frac{15}{2}$ and $b=12$.

Step 2: Find the equation of straight line

The intercept form of a straight line can be given by $\frac{x}{a}+\frac{y}{b}=1$.

Substitute $a$ with $\left(-\frac{15}{2}\right)$ and $b$ with $12$.

$$\begin{gathered} \Rightarrow \frac{x}{\left(-{\displaystyle \frac{15}{2}}\right)}+\frac{y}{12}=1 \\ \Rightarrow -\frac{2x}{15}+\frac{y}{12}=1 \\ \Rightarrow \frac{-\left(2\times 4\right)x+5y}{60}=1 \\ \Rightarrow -8x+5y=60 \\ \Rightarrow 8x-5y+60=0 \end{gathered}$$

Therefore, the equation of the straight line is $8x-5y+60=0$.