Question

Find the coordinates of the point which divides the line segment joining $( - 1,3)$ and $(4, - 7)$ internally in the ratio $3:4.$

Answer 1

Hint-In order to solve such a question we will simply use the section formula which tells us the coordinates of the point which divides a given line segment into two parts such that their length is in the ratio m:n.
$\left[ {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right]$
Where m, n is the ratios and x, y is the coordinates.

Complete step-by-step answer:

We will solve the problem with the help of a given figure.
As we know that section formula or required coordinates of the point is given as
$\left[ {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right]$
Given ratio is $3:4$ and the coordinates of the line are $( - 1,3)$ and $(4, - 7)$ .
So, here $m = 3,n = 4$ and ${x_1} = - 1,{y_1} = 3,{x_2} = 4,{y_2} = - 2$
Substituting these values in the section formula given above, we get

$$\Rightarrow \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right) \\\ \Rightarrow \left( {\dfrac{{3 \times 4 + 4 \times ( - 1)}}{{3 + 4}},\dfrac{{3 \times ( - 7) + 4 \times 3}}{{3 + 4}}} \right) \\\ \Rightarrow \left( {\dfrac{{12 - 4}}{7},\dfrac{{ - 21 + 12}}{7}} \right) \\\ \Rightarrow \left( {\dfrac{8}{7},\dfrac{{ - 9}}{7}} \right) \\\$$

Hence, the coordinates of the point which divides the line segment joining $( - 1,3)$ and $(4, - 7)$ internally in the ratio $3:4$is $\left( {\dfrac{8}{7},\dfrac{{ - 9}}{7}} \right)$.

Note- To solve these types of problems remember all the formulas of coordinate geometry. And try to draw a rough sketch of the diagram on the paper, this helps a lot in solving the question. Graphs method is always the easiest and least time consuming method.