1. The problem statement, all variables and given/known data A circle with radius 1 inscribed in the parabola y=x^2. Find the center of the circle. 2. Relevant equations equation of the parabola: y=x^2 circle: (x-h)^2 + (y-k)^2 = 1 3. The attempt at a solution After ghosting the forums and reading through every post with this exact same question, I can't quite get to the answer, which is (0,5/4). i've assigned notation to the coordinates of the center of the circle C(h,k), and the the two points where it intersects the parabola P(x1,(f(x1)), and Q(-x1,f(-x1)). where f(x) functions y: y=f(x)=x^2. since the center of the circle lies on the y-axis, therefore h=0. thus, the new equation of the circle is: x^2 + (y-k)^2 = 1 by relating the parabola to the equation of the circle, i get: x^2 + (x^2-k)^2=1 this is where i am a bit confused about how to proceed to find the value of k. i just finished calculus 1 (differentiation) this spring quarter and can't for the life of me think of how to solve this problem which is quite infuriating. from reading other posts here, i've managed to arrive at what various responses to this same question have instructed me to do, but am still lost as how finish it... here's what i got. from differentiating the equation of the circle, i get: 2x + 2(x^2-k)(2x-0)=0 -> 2x + 4x(x^2-k)=0 -> 4x(x^2-k)=-2x -> 1 = -(1/2(x^2-k)) -> 2(x^2-k) = -1 i've also tried expanding the equation before differentiating, i get x^2 + x^4 -2x^2k + k^2 = 1 from grouping like-terms and factoring x^2, i get: x^4 + (1-2k)x^2 + k^2 = 1 and here i am again am stumped as how to proceed... i've tried using the quadratic equation where a=1, b=(1-2k) (?) and c=1 but it that doesn't seem right either.