How to Make a Circle Tangent to a Parabola at a Given Point?

[dmonlama]

An Osculating Circle! HELPPP plZZ urgent

Homework Statement

find the values of h,k and a that make the circle (x-h)^2+(y-k)^2=a^2 tangent to the parabola y=x^2+1 at the point (1,2) annd that also make the second derivatives d^2y/dx^2 have the same value on both courves.

 

[HallsofIvy]

What have YOU done? You need to find three values, h, k, and a, and you have three pieces of information: the circle passes through (1, 2):$(1-h)^2+ (2-k)^2= a^2$; The derivative of $(x-h)^2+ (y-k)^2= a^2$ at (1, 2) is the same as the derivative of $y= x^2+ 1$ at (1, 2); the second derivative of $(x-h)^2+ (y-k)^2= a^2$ at (1, 2) is the same as the second derivative of $y= x^2+1$ at (1, 2). The first and second derivatives of $y= x^2+ 1$ at (1,2) are easy. I would recommend using "implicit differentiation" to find the derivatives of $(x-h)^2+ (y-k)^2= a^2$.