Chapter 12.5, Problem 55ES

These exercises are concerned with the problem of creating a single smooth curve by piecing together two separate smooth curves. If two smooth curves $C_1\text{ and }{C}_2$ are joined at a point P to form a curve C, then we will say that $C_1\text{ and }{C}_2$ make a smooth transition at P if the curvature of C is continuous at P.

The accompanying figure shows the arc of a circle of radius r with center at $\left(0,r\right).$ Find the value of a so that there is a smooth transition from the circle to the parabola $y=a{x}^2$ at the point where $x=0.$