Find the equation of a circle centre on the line y=2x+1 touching the y axis and passing through A(4,5)

Answer 1

a general circle is

(x-h)^2 + (y-k)^2 = r^2

since the center is on y=2x+1, k=2h+1

(x-h)^2 + (y-2h-1)^2 = r^2
Since the circle passes through (4,5),

(4-h)^2 + (5-2h-1)^2 = r^2

Since the circle is tangent to the y-axis, h=r, so

(4-r)^2 + (4-2r)^2 = r^2
r = 2 or 4

You can now check to see whether those are both possible solutions, after substituting back to get (h,k).