Question

(-6,2) & (3,5) points are divided by point (2,5) find the ratio​

Answer 1

Given: The line formed by the points (-6, 2) and (3, 5) are divided by the point (2, 5).

To find: The ratio in which it is done so.

Answer:

Section formula:

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

Let the ratio be k:1.

From the given data, we have,

$\tt m\ =\ k\\\\n\ =\ 1\\\\x_1\ =\ -6\\\\x_2\ =\ 3\\\\y_1\ =\ 2\\\\y_2\ =\ 5$

Using them in the formula,

$\tt \Bigg(2,\ 5\Bigg)\ =\ \Bigg(\dfrac{(k\ \times\ 3)\ +\ (1\ \times\ -6)}{k\ +\ 1},\ \dfrac{(k\ \times\ 5)\ +\ (1\ \times\ 2)}{k\ +\ 1}\Bigg)\\\\\\\Bigg(2,\ 5\Bigg)\ =\ \Bigg(\dfrac{3k\ -\ 6}{k\ +\ 1},\ \dfrac{5k\ +\ 2}{k\ +\ 1}\Bigg)$

Equating the x - coordinates, (same can be done with the y - coordinates as well)

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

Therefore, the line formed by the points (-6, 2) and (3, 5) are divided by the point (2, 5) in the ratio 8:1.