Formula to find the coordinates of a point using section formula concept is as follows:

The line Segment joining points A(x_{1}, y_{1}) and B(x_{2}, y_{2}), is divided by the point P which has the coordinates (x, y).

- The coordinates of the point dividing the line segment joining
**(x**and_{1}, y_{1})**(x**in the ratio_{2}, y_{2})**m1 : m2 internally**is given by ( (*m*2+_{1}x*m*1)_{2}x**/**(*m*+_{1}*m*,(_{2})*m*2+_{1}y*m*1)_{2}y**/**(*m*+m_{1}_{2}*)*) - The coordinates of the point dividing the line segment joining
**(x**and_{1}, y_{1})**(x**in the ratio_{2}, y_{2})**m1 : m2 externally**is given by ( (*m*2 -_{1}x*m*1)_{2}x**/**(*m*-_{1}*m*,(_{2})*m*2 -_{1}y*m*1)_{2}y**/**(*m*m_{1 - }_{2}*)*)

Find the coordinates of the point which divides the line segment joining the points (4, – 3) and (8, 5) in the ratio 3 : 1 internally

**Solution:** Using section formula P(x, y) = { (m_{1}x_{2} + m_{2}x_{1})/(m_{1} + m_{2} ),(m_{1}y_{2} + m_{2}y_{1})/(m_{1} + m_{2} ) }

We get P(x, y) = { (3 X 8 + 1 X 4 )/(3+1),(3 X 5 + 1 X -3)/(3+1) } = (7,3)

Find the coordinates of the point which divides AB, where A and B have the coordinates (2,4) and (4,6) respectively, in the ratio 1:3 (i) internally (ii) externally

The figure illustrates the two cases. Notice that in both cases AC: CB=1:3

We just have to put in the values in the section formula mentioned above

1. x= ((1×4)+(3×2)) / (1+3)= 5 / 2 and y=( (1×6)+(3×4) )/ (1+3)= 9 / 2 .

(a) Therefore the coordinates of the required point are (5/2,9/2)

2. (ii) x=((1×4)−(3×2) )/ (1−3)=1 and y= ( (1×6)−(3×4) ) / (1−3) =3

Therefore the coordinates of the required point are (1,3)

We need to ensure that we take care of the order of A and B. It is suggested to construct the points and figure to avoid any mistakes in misinterpreting and avoid arriving at erroneous calculations.