Chapter 3, Problem 5AAE

An osculating circle Find the values of $h,k,$ and $a$ that make the circle ${\left(x-h\right)}^2+{\left(y-k\right)}^2=a^2$ tangent to the parabola $y=x^2+1$ at the point $\left(1,2\right)$ and that also make the second derivatives $d^{2}y/d{x}^2$ have the same value on both curves there. Circles like this one that are tangent to a curve and have the same second derivative as the curve at the point of tangency are called osculating circles (from the Latin osculari, meaning “to kiss”). We will encounter them again in Chapter 13.