Question

In the figure given above, the line segment $AB$ meets $X$-axis at $A$ and $Y$-axis at $B$. The point $P(-3,4)$ on $AB$ divides it in the ratio $2:3$. 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 the co-ordinates of $A$ and $B$ be $(x,0)\equiv (x_1,y_1)$ and $(0,y)\equiv (x_2,y_2)$

$\because$ The co-ordinates of a point $P(-3,4)$ on AB divides it in the ratio $2:3$.

i.e., $AP:PB=2:3$

$\therefore$ $m=2$ and $n=3$

and $x=-3$ and $y=4$

By using section formula, we get

$-3=\frac{2\times 0+3\times x}{2+3}$

$-3=\frac{3x}{5}\Rightarrow 3x=-15$

$\Rightarrow x=-5$

and $4=\frac{2\times y+3\times 0}{2+3}$

$4=\frac{2y}{5}\Rightarrow 2y=20$

$\Rightarrow y=10$.

Hence, the co-ordinates of $A$ and $B$ are $(-5,0)$ and $(0,10)$.