Question

Find the coordinates of the point P which divides the line segment joining A(4,3) and B(8,5) in the ratio 2:1 internally.

Answer 1

$\left(\frac{20}{3}, \frac{13}{3}\right)$

Answer 2

The coordinates of the point P which divides the line segment joining A($x_1, y_1$) and B($x_2, y_2$) in the ratio m:n internally are given by the section formula:

$P(x, y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)$

Given:
Point A($x_1, y_1$) = (4, 3)
Point B($x_2, y_2$) = (8, 5)
Ratio m:n = 2:1 (so, m = 2, n = 1)

Substitute the values into the formula:

For the x-coordinate of P:
$x = \frac{2 \times 8 + 1 \times 4}{2+1}$
$x = \frac{16 + 4}{3}$
$x = \frac{20}{3}$

For the y-coordinate of P:
$y = \frac{2 \times 5 + 1 \times 3}{2+1}$
$y = \frac{10 + 3}{3}$
$y = \frac{13}{3}$

Therefore, the coordinates of point P are $\left( \frac{20}{3}, \frac{13}{3} \right)$.