Question

Find the ratio in which point P(–1, m) divides the line segment joining the points A(2, 5) and В(–5, –2). Hence, find the value of m.

Answer 1

Let the ratio in which point P(–1, m) divides the line segment be 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(–1, m) = `((-5k + 2)/(k + 1), (-2k + 5)/(k + 1))`

On comparing,

∴ `(-5k + 2)/(k + 1) = -1`   ...(1)

`(-2k + 5)/(k + 1) = m`   ...(2)

From equation (1)

`(-5k + 2)/(k + 1) = -1`

⇒ –5k + 2 = –1(k + 1)   ...(By cross multiplying)

⇒ –5k + 2 = –k – 1

⇒ –5k + k = – 1 – 2

⇒ –4k = –3

⇒ k = `3/4`

So, the ratio in which p divides the line segment is `3/4`, i.e., 3 : 4.

Now, putting the value of k in equation (2),

`(-2k + 5)/(k + 1) = m`

⇒ `(-2(3/4) + 5)/(3/4 + 1) = m`

⇒ `((-3)/2 + 5)/((3  +  4)/4) = m`

⇒ `((-3  +  10)/2)/(7/4) = m`

⇒ `7/2 xx 4/7 = m`

⇒ m = 2

Therefore, the value of m is 2.