Coordinates

Find all points (x,y) that are 5 units away from the point (2,7) and that lie on the line y = 5x - 28

If we plug it into desmos and draw a circle of \(\left(x-2\right)^{2}+\left(y-7\right)^{2}=25\), then we can see all the points of intersection, and there are 2.

\((7,7)\)

\((6.615,5.077)\) That is not the value, it is just an approximation.

We want to find the intersection points of

(x -2)^2  + (y -7)^2   = 25       and  y = 5x -28

So

(x -2)^2  + (5x -28 - 7)^2   =   25

(x -2)^2  + (5(x -7))^2    = 25

(x -2)^2 + 25(x - 7)^2  = 25

x^2 - 4x + 4 + 25 (x^2 - 14x + 49)  = 25

x^2  - 4x + 4  + 25x^2 - 350x + 1225 - 25 =  0

26x^2 - 354x +  1204   =  0

x =  (354 + sqrt [ 354^2  - 4*26*1204 ])   / 52    =  7    and

x =   (354 - sqrt [ 354^2  - 4*26*1204 ])   / 52 =   86/13

So   y = 5(7) - 28 =  7

And

y = 5(86/13) - 28  =  66 /13

So  the points  are   (7,7)   and (86/13  , 66 / 13)