A, B, and C are three collinear points. The coordinates of A and B are (3, 4) and (7, 7) respectively and AC =10 units. Find sum of co-ordinates of C.

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Solution

A (3,4) and B(7,7) and C(x,y) or C'(x,y) are collinear points.
AB = √[(7-3)²+(7-4)²] = 5
AC = 10, given.

Slope of AC = slope of AB = (7-4)/(7-3) = 3/4
=> (y-4)/(x-3) = 3/4 --- (1)
=> 4 y - 3 x = 7 --- (2)

AC² = 10² = (y - 4)² + (x - 3)²
= [ 3/4 * (x - 3) ]² + (x-3)²
= (x-3)² * 25/16
=> x - 3 = + 8
=> x = +11 or -5

=> y = (7+3x)/4
= 10 or -2

C = (11, 10) and C' = (-5, -2)

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