Question

Find the ratio in which the point 2, 5 divides the line segment joining 8, 2 and minus -6, 9

Answer 1

Given: (2, 5) divides the line segment joining the points (8, 2) and (-6, 9).

To find: The ratio in which it is divided.

Answer:

Let the ratio be k:1.

Section formula:

$\tt \bigg(\dfrac{mx_2\ +\ nx_1}{m\ +\ n},\ \dfrac{my_2\ +\ ny_1}{m\ +\ n}\bigg)$

From the given data, we have:

$\tt x_1\ =\ 8\\\\x_2\ =\ -6\\\\y_1\ =\ 2\\\\y_2\ =\ 9$

Using them in the formula,

$\tt\\\\(2,\ 5)\ =\ \bigg(\dfrac{(k\ \times\ -6)\ +\ (1\ \times\ 8)}{k\ +\ 1},\ \dfrac{(k\ \times\ 9)\ +\ (1\ \times\ 2)}{k\ +\ 1}\bigg)\\\\\\(2,\ 5)\ =\ \bigg(\dfrac{-6k\ +\ 8}{k\ +\ 1},\ \dfrac{9k\ +\ 2}{k\ +\ 1}\bigg)$

Equating the x coordinates, [same can be done with the y coordinates as well.]

$\tt 2\ =\ \dfrac{-6k\ +\ 8}{k\ +\ 1}\\\\\\2k\ +\ 2\ =\ -6k\ +\ 8\\\\\\2k\ +\ 6k\ =\ 8\ -\ 2\\\\\\8k\ =\ 6\\\\\\k\ =\ \dfrac{6}{8}\\\\\\k\ =\ \dfrac{3}{4}$

Therefore, the ratio is 3:4.

Hence, (2, 5) divides the line segment joining the points (8, 2) and (-6, 9) in that ratio 3:4.