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?
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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 \( \left(14, 19\right) \).