Question

$A(-3,4)$ and $B(2,1)$ be any two given point. If $C$ be a point on AB produced such that $AC=2BC$, then the coordinates of C are ?

Answer 1

We know that by section formula, the co-ordinates of the points which divide internally the line segment joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ is

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

Since coordinate of $A=(-3,4)$

and $B=(2,1)$

also, C be a point produced on $AB$

such that, $AC=2BC\Rightarrow \frac{AC}{BC}=\frac{2}{1}\Rightarrow m:n=2:1$

Let the coordinate of $C$ is $(x,y)$

Hence,

$x=\frac{m{x}_2+n{x}_1}{m+n}=\frac{2.2+1.(-3)}{2+1}=\frac{1}{3}y=\frac{m{y}_2+n{y}_1}{m+n}=\frac{2.1+1.4}{2+1}=2$

therefore the coordinate of $C$ is $(\frac{1}{3},2)$