Minimum distance between any two curves lie along common normal

So if we find the coordinate where common normal touches the parabola, then that coordinate will be closest to circle

y^2 = 8x

x^2 + (y+6)^2 = 1

Equation normal to parabola:

y = mx - 2(2)m-2m^3

where m is slope of normal

y = mx - 4m-2m^3

So this common normal will pass through the center of circle also

Center (0, -6)

So, -6 = m(0) - 4m-2m^3

m^3 + 2m - 3 = 0

m = 1 is one solution

y = x - 6

Put this in parabola

y^2 = 8(y+6)

y^2 - 8y - 48 = 0

y = -4, 12

x = 2, 18

**(2, -4)** is the closest point