Question

Find the coordinates of the point which divides the line joining the points \[(4, - 3)\] and $(8,5)$ in the ratio $3:1$ internally.

Answer 1

Hint: There are equations for finding the coordinates of a point which divides a line segment called section formula. Here the endpoints of the segment and the ratio are given. Using this we can find the coordinates.

Formula used: The coordinates of a point dividing a line segment $AB$ in the ratio $l:m$, where $A({x_1},{y_1})$ and $B({x_2},{y_2})$ are $(\dfrac{{l{x_2} + m{x_1}}}{{l + m}},\dfrac{{l{y_2} + m{y_1}}}{{l + m}})$.

Complete step-by-step answer:
Given the points \[(4, - 3)\] and $(8,5)$.
We have to find the coordinates of the point which divides the line joining the points \[(4, - 3)\] and $(8,5)$ in the ratio $3:1$ internally.
Let the points be $A$ and $B$.
The coordinates of a point dividing a line segment $AB$ in the ratio $l:m$, where $A({x_1},{y_1})$ and $B({x_2},{y_2})$ are $(\dfrac{{l{x_2} + m{x_1}}}{{l + m}},\dfrac{{l{y_2} + m{y_1}}}{{l + m}})$.
Here, $l = 3,m = 1$
Also ${x_1} = 4,{x_2} = 5,{y_1} = - 3.{y_2} = 5$
So substituting we get,
The coordinates of a point dividing a line segment $AB$ in the ratio $3:1$, where $A(4, - 3)$ and $B(8,5)$ are $(\dfrac{{3 \times 8 + 1 \times 4}}{{3 + 1}},\dfrac{{3 \times 5 + 1 \times - 3}}{{3 + 1}})$.
Simplifying we get,
The coordinates are $(\dfrac{{24 + 4}}{4},\dfrac{{15 + - 3}}{4}) = (\dfrac{{28}}{4},\dfrac{{12}}{4}) = (7,3)$

Therefore, the answer is $(7,3)$.

Note: This formula is called section formula. Here we used the case of internal division. We have another formula for external division as well. The coordinates of a point that lie outside the line, where the ratio of the length of a point from both the line segments are in the ratio $l:m$ is $(\dfrac{{l{x_2} - m{x_1}}}{{l - m}},\dfrac{{l{y_2} - m{y_1}}}{{l - m}})$. The only difference here is the sign in between. So we have to be careful in using the formula. We must use the addition sign for internal division and subtraction for external division.