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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.
Solution
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.
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