What are the coordinates of point $${B}$$ on $$A C$$ such that the ratio of $$A B$$ to $$A C$$ is $$1: 3$$ ? ![](http://admin.getprephub.com/media/editor/ag12_20230214204144147301.png)

Here, $$A(2,7)$$ and $$C(-2,-3)$$ are the coordinate of $$A$$ and $$C$$ respectively. $$B$$ divides $$AC$$ in the ratio $$1: 2$$. since, $$\frac{A B}{A C}=\frac{1}{3}$$ $$\text { or, } \frac{A B+B C}{A B} =\frac{3}{1} $$ $$\text { or, } 1+\frac{B C}{A B} =1+\frac{2}{1}$$ $$\therefore \frac{A B}{B C}=\frac{1}{2}$$ By section formula, $$ B(x, y)=(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}) $$ $$ m=1, n=2, x_1=2, x_2=-2, y_1=7, y_2=-3 $$ $$ B(x, y)=(\frac{1 \cdot(-2)+2,2}{2+1}, \frac{1 \cdot(-3)+2 \cdot 7}{1+2}) $$ $$ =(\frac{-2+4}{3}, \frac{-3+14}{3}) $$ $$ =(\frac{2}{3}, \frac{11}{3})=(\frac{2}{3}, 3 \frac{2}{3}) $$

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MCQ

What are the coordinates of point ${B}$ on $A C$ such that the ratio of $A B$ to $A C$ is $1: 3$ ?

Select all that apply :

$(\frac{2}{3}, 3 \frac{2}{3})$

$(\frac{1}{3}, 2 \frac{5}{6})$

$(-\frac{1}{3}, 1 \frac{1}{6})$

$(-\frac{2}{3}, \frac{1}{3})$

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