Question

Find the ratio in which the segment joining the points (2, –5) and (5, 3) is divided by the x-axis. Also, find the coordinates of the point on the x-axis.

Answer 1

Let the point on the x-axis be (x, 0) since the y-coordinate is always zero on the x-axis.

Let point P(x, 0) divide the line segment joining the points A(2, –5) and B(5, 3) in the ratio k : 1.

Then, coordinates of P = `((m_1x_2 + m_2x_1)/(m_1 + m_2), (m_1y_2 + m_2y_1)/(m_1 + m_2))`

⇒ P(x, 0) = `((5k + 2)/(k + 1), (3k - 5)/(k + 1))`

∴ `(5k + 2)/(k + 1) = x`   ...(1)

And `(3k - 5)/(k + 1) = 0`   ...(2)

3k – 5 = 0

⇒ 3k = 5

⇒ k = `5/3`

Putting the value of k in equation (1),

`(5(5/3) + 2)/(5/3 + 1) = x`

⇒ `(25/3 + 2)/((5  +  3)/3) = x`

⇒ `((25  +  6)/3)/(8/3) = x`

⇒ `31/3 xx 3/8 = x`

⇒ x = `31/8`

Therefore, the coordinates of the point `P(31/8, 0)` on the x-axis and ratio is 5 : 3, which is the segment AB divided by x-axis.