this is a calculus-geometry question that I found -
A circle of radius $1$ is tangent to the parabola $y = x^2 -2x + 1$ at two points. Find the coordinates of the center of the circle.
Let the center of the circle be $(a, b)$. Since there are two intersection points, we have:
$(x - a)^2 + (y - b)^2 - 1 = x^2 - 2x + 1$
$x^2 - 2ax + a^2 + y^2 - 2ay + b^2 = x^2 - 2x$
$-2ax + a^2 + y^2 - 2ay + b^2 = -2x$
I am unsure of how to proceed; I think I am approaching this problem wrong because I did not take the derivative. The correct answer is $(1, 5/4).$