Question

5. (a) Find the ratio in which X-axis divides the line segment joining the points (2, -3)
and (5, 6). Then find the intersecting point on X-axis.​

Answer 1

Given: Points (2, -3) and (5, 6) being divided by the x - axis.

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

Answer:

Section formula:

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

Let's assume the ratio is k:1.

From the given data, we have:

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

Since it's divided by the x - axis, the point will be (x, 0).

Using the values we have in the formula,

$\tt (x, 0)\ =\ \bigg(\dfrac{(k\ \times\ 5)\ +\ (1\ \times\ 2)}{k\ +\ 1},\ \dfrac{(k\ \times\ 6)\ +\ (1\ \times\ -3)}{k\ +\ 1}\bigg)$

Equating the y coordinates,

$\tt 0\ =\ \dfrac{6k\ -\ 3}{k\ +\ 1}\\\\\\0\ =\ 6k\ -\ 3\\\\\\3\ =\ 6k\\\\\\\dfrac{1}{2}\ =\ k$

Therefore, the ratio is 1:2.

Now, equating the x coordinates,

$\tt x\ =\ \dfrac{5k\ +\ 2}{k\ +\ 1}\\\\\\x(k\ +\ 1)\ =\ 5k\ +\ 2\\\\\\x\bigg(\dfrac{1}{2}\ +\ 1\bigg)\ =\ \dfrac{5}{2}\ +\ 2\\\\\\x\bigg(\dfrac{3}{2}\bigg)\ =\ \dfrac{9}{2}\\\\\\x\ =\ \dfrac{9}{2}\ \times\ \dfrac{2}{3}\\\\\\x\ =\ 3$

Therefore, the intersecting point on the x - axis is (3, 0).