Find the coordinates of the point which divides the join of (−1, 7) and (4, −3) in the ratio 2 : 3.
(1, 3)
We use the section formula to find the point dividing the segment between two points in a given ratio.
The given points are \(A(-1, 7)\) and \(B(4, -3)\), and the ratio is \(2 : 3\).
For internal division, the coordinates of the point \(P(x, y)\) dividing \(AB\) in the ratio \(m : n\) are:
\( x = \frac{mx_2 + nx_1}{m + n} \) and \( y = \frac{my_2 + ny_1}{m + n} \).
Here, \(m = 2\), \(n = 3\), \(x_1 = -1\), \(y_1 = 7\), \(x_2 = 4\), \(y_2 = -3\).
Substitute in the formula:
\( x = \frac{2(4) + 3(-1)}{2 + 3} = \frac{8 - 3}{5} = \frac{5}{5} = 1 \).
Similarly,
\( y = \frac{2(-3) + 3(7)}{2 + 3} = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \).
So the required point is \((1, 3)\).