Question

If A , B , C are collinear points such that A = ( 3 , 4 ) , B = ( 7 , 7 ) and AC = 10 , then C = ?

Answer 1

see diagram.

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      using (1)
                 = (x-3)² * 25/16
=>  x - 3 = + 8
=>  x = +11  or  -5 

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

C = (11, 10)   and   C' = (-5, -2)

Answer 2

Given:-

✬ A, B and C are three collinear points, where
A = (3,4) and B(7 ,7)

we know distance formula......

D = S/T

so,

Distance between A and B, we get
=

and
Distance between A and C is 10units
so,

Distance between B and C = 10-5= 5 units....

Let coordinate of "c" = (x, y)

so, we get
and

Now we subtract equation 2 from equation 1,we get.

8x+6y = 148
4x+3y = 74
4x= 74-3y
x=, substitute that value in eq 1, we get

so,
x =
so,
coordinate of C = (11, 10)