22. Find out the co-ordinates of the points which divides internally the line joining the point \( (2,1) \) and \( (6,7) \) in the ratio( \( 3: 2) \). What will be the co-ordinate if it is divided externally?

To find the coordinates of the point that divides the line segment joining \( (2, 1) \) and \( (6, 7) \) in the ratio \( 3:2 \) internally, you can use the section formula. The formula for the coordinates \( (x, y) \) of the dividing point is given by: \[ x = \frac{m x_2 + n x_1}{m + n} \quad \text{and} \quad y = \frac{m y_2 + n y_1}{m + n} \] where \( m \) and \( n \) are the parts of the ratio and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the points. Here, \( (x_1, y_1) = (2, 1) \), \( (x_2, y_2) = (6, 7) \), \( m = 3 \), and \( n = 2 \). Calculating: \[ x = \frac{3 \cdot 6 + 2 \cdot 2}{3 + 2} = \frac{18 + 4}{5} = \frac{22}{5} = 4.4 \] \[ y = \frac{3 \cdot 7 + 2 \cdot 1}{3 + 2} = \frac{21 + 2}{5} = \frac{23}{5} = 4.6 \] So, the internal division gives the point \( \left(4.4, 4.6\right) \). For the external division in the ratio \( 3:2 \), you use the same formula, but with a negative sign for the ratio's second term: \[ x = \frac{m x_2 - n x_1}{m - n} \quad \text{and} \quad y = \frac{m y_2 - n y_1}{m - n} \] Calculating: \[ x = \frac{3 \cdot 6 - 2 \cdot 2}{3 - 2} = \frac{18 - 4}{1} = 14 \] \[ y = \frac{3 \cdot 7 - 2 \cdot 1}{3 - 2} = \frac{21 - 2}{1} = 19 \] Thus, the external division gives the point \( \left(14, 19\right) \).