Question

Find the ratio in which the point `P(3/4, 5/12)` divides the line segment joining the points `A(1/2, 3/2)` and B(2, –5). 

Answer 1

Let `"P"(3/4, 5/12)` divide AB internally in the ratio m : n

Using the section formula, we get

`(3/4, 5/12) = ((2m - n/2)/(m + n), (-5m + 3/2n)/(m + n))`   ...`[∵ "Internal section formula, the coordinates of point P divides the line segment joining the point"  (x_1, y_1)  "and"  (x_2, y_2)  "in the ratio"  m_1 : m_2  "internally is"  ((m_2x_1 + m_1x_2)/(m_1 + m_2), (m_2y_1 + m_1y_2)/(m_1 + m_2))]`

On equating, we get

`3/4 = (2m - n/2)/(m + n)` and `5/12 = (-5m + 3/2n)/(m + n)`

⇒ `3/4 = (4m - n)/(2(m + n))` and `5/12 = (-10m + 3n)/(2(m + n))`

⇒ `3/2 = (4m - n)/(m + n)` and `5/6 = (-10m + 3n)/(m + n)`

⇒ 3m + 3n = 8m – 2n and 5m + 5n = – 60m + 18n

⇒ 5n – 5m = 0 and 65m – 13n = 0

⇒ n = m and 13(5m – n) = 0

⇒ n = m and 5m – n = 0

Since, m = n does not satisfy.

∴ 5m – n = 0

⇒ 5m = n

∴ `"m"/"n" = 1/5`

Hence, the required ratio is 1 : 5.