The intersection points will be where the line with the equation ( x - 3)^2 + ( y + 8)^2 = 12^2 intersects the line y = 2x +1
Sub the equation of the line into the equation of the circle for y and we have
(x - 3)^2 + [ (2x + 1) + 8 ]^2 = 12^2 simplify
x^2 - 6x + 9 + ( 2x + 9)^2 =144
x^2 - 6x + 9 + 4x^2 + 36x + 81 = 144
5x^2 + 30x + 90 = 144 subtract 144 from both sides
5x^2 + 30x - 54 = 0
x = -30 + sqrt ( 900 + 20 *54) -30 + sqrt (1980) -30 + 6sqrt (55)
______________________ = ________________ = ________________ =
2 * 5 10 10
[-15 + 3sqrt ( 55)] / 5
And the other x value is [-15 -3sqrt (55) ] / 5
So y = (2/5) ( -15 + 3sqrt (55)) + 1 = -6 + (6/5)sqrt (55) +1 = -5 + (6/5)sqrt (55)
And the other y = -5 - (6/5)sqrt (55)
So the intersection points are
( [ -15 + 3sqrt (55)] / 5, -5 + (6/5)sqrt (55) ) and ( [ -15 -3sqrt (55) ] / 5 , -5 - (6/5)sqrt (55) ]