Question

A line $AB$ meets X-axis at $A$ and Y-axis at $B.P(4,-1)$ divides $AB$ in the ratio $1:2$.
Find the coordinates of $A$ and $B$

Answer 1

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Let coordinates of $A$ and $B$ be $(a,0)$ and $(0,b)$
$\because$ The coordinate of a point $P(4,-1)$ on $AB$ divides it in the ratio $1:2$.
i.e., $AP:PB=1:2$
By using section formula,
$4=\frac{1\times 0+2\times a}{1+2}[\because x=\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2}]$
$12=2a\Rightarrow a=6$
and $-1=\frac{1\times b+2\times 0}{1+2}[\because y=\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2}]$
$\Rightarrow -3=b$
Hence, Coordinate of $A=(6,0)$
and Coordinate of $B=(0,-3)$