Question

If P(9a – 2, – b) divides line segment joining A(3a + 1, –3) and B(8a, 5) in the ratio 3 : 1, find the values of a and b.

Answer 1

Let P(9a – 2, – b) divides AB internally in the ratio 3 : 1.

By section formula,

9a – 2 = `(3(8a) + 1(3a + 1))/(3 + 1)`  ...`[∵ "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))]`

And – b = `(3(5) + 1(-3))/(3 + 1)`

⇒ 9a – 2 = `(24a + 3a + 1)/4`

And – b = `(15 - 3)/4`

⇒ 9a – 2 = `(27a + 1)/4`

And – b = `12/4`

⇒ 36a – 8 = 27a + 1

And b = – 3

⇒ 36a – 27a – 8 – 1 = 0

⇒ 9a – 9 = 0

∴ a = 1

Hence, the required values of a and b are 1 and – 3.