Question

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

• A. (10/3, 11/3)
• B. (11/3, 10/3)
• C. (5/3, 10/3)
• D. (5/3, 7/3)

Answer 1

The calculated answer (3,4) is not among the given options.

Answer 2

To find the coordinates of point P which divides the line segment joining A(1,2) and B(4,5) in the ratio 2:1 internally, we use the section formula for internal division.

Section Formula:

If a point P(x, y) divides the line segment joining A($x_1, y_1$) and B($x_2, y_2$) in the ratio m:n internally, then its coordinates are given by:

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

Given values:

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

Calculate the x-coordinate of P:

$x = \frac{(2 \times 4) + (1 \times 1)}{2+1}$

$x = \frac{8 + 1}{3}$

$x = \frac{9}{3}$

$x = 3$

Calculate the y-coordinate of P:

$y = \frac{(2 \times 5) + (1 \times 2)}{2+1}$

$y = \frac{10 + 2}{3}$

$y = \frac{12}{3}$

$y = 4$

Therefore, the coordinates of point P are (3, 4).

Upon comparing this result with the given options:

(10/3, 11/3) (11/3, 10/3) (5/3, 10/3) (5/3, 7/3)

The calculated coordinates (3, 4) or (9/3, 12/3) do not match any of the provided options. This indicates that there might be an error in the options provided for the question. However, based on the problem statement and correct application of the section formula, (3,4) is the correct answer.