Question

Point P(x, 0) divides the line segment joining the points (2, 8) and (–3, –5) in a certain ratio. Find the ratio and hence find the value of x.

Answer 1

Let the ratio in which point P(x, 0) divides the line segment be k : 1.

Then, the coordinate 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) = `(-3k + 2)/(k + 1), (-5k + 8)/(k + 1)`

∴ `(-3k + 2)/(k + 1) = x`   ...(1)

And `(-5k + 8)/(k + 1) = 0`   ...(2)

⇒ – 5k + 8 = 0   ...[From (2)]

⇒ `k = (-8)/(-5)`

⇒ `k = 8/5`

Putting the value of k in equation (1), we have

`(-3(8/5) + 2)/(8/5 + 1) = x`

⇒ `((-24)/5 + 2)/((8  +  5)/5) = x`

⇒ `((-24  +  10)/5)/(13/5) = x`

⇒ `(-14)/5 xx 5/13 = x`

⇒ `x = (-14)/13`

Hence, ratio = 8 : 5 and x = `(-14)/13`.