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

Given that, P divides the line segment AB in the ratio of 3: 1.

A (3a+1 , -2) = (x1, y1)

B (8a, 5) = (x2, y2)

m + n = 4

By using the section formula, 

x = (mx2 + bx1)/m+n and y = (my2+ny1)/(m+n)

Now, substituting the known values in the formula, we get

x = (3(8a) + 1(3a+1))/4 and y = (3(5)+1(-2))/4

x = (24a + 3a +1)/4 and y = (15-2)/4

Finding the value of a:

9a – 2 = (27a+ 1)/4

36a – 8 = 27a + 1

9a = 9

a = 1.

Finding the value of b:

-b = 13/4

b = -13/4

Hence, the value of a and b are 1 and -13/4, respectively.

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