Question

Equation of the line which passes through the point $(-4,3)$ and the portion of the line intercepted between the axes is divided internally in the ratio $5:3$ by this point, is

• A. 9x+20y+96=0
• B. 20x+9y+96=0
• C. 9x-20y+96=0
• D. None of these

Answer 1

The correct option is B ($9x-20y+96=0$)

Let the line cut the x-axis at $A(a,0)$ and y-axis $B(0,b)$
It is given that the point $(-4,3)$ divides the line $AB$ internally in the ratio of $5:3$
By the section formula, $x=\frac{m{x}_2+n{x}_1}{m+n}$ and $y=\frac{m{y}_2+n{y}_1}{m+n}$

Where $x_1=a,y_1=0,x_2=0,y_2=b,x=-4,y=3$ and $m=5,n=3$

$\Rightarrow -4=\frac{(5\times 0)+(3\times a)}{5+3}$ and $3=\frac{(5\times b)+(3\times 0)}{5+3}$

$\Rightarrow -4=\frac{3a}{8}$ and $y=\frac{5b}{8}$

$\therefore a=\frac{-32}{3}$ and $b=\frac{24}{5}$

Thus, the required equation in intercept form is $\frac{x}{a}+\frac{y}{b}=1$
$\Rightarrow \frac{x}{\frac{-32}{3}}+\frac{y}{\frac{24}{5}}=1$

$\Rightarrow -\frac{3x}{32}+\frac{5y}{24}x=1$

$\Rightarrow \frac{-9x+20y}{96}=1$

$\Rightarrow -9x+20y=96$

$\therefore 9x-20y+96=0$ is the required equation.