Curvature — Question 6

Question 6 – Solution

Strategy Use curvature for the radius and the principal normal for the center direction.

Curvature and normal From Question 2, $\kappa(0)=2$, so $\rho=1/\kappa=1/2$. At the vertex, the unit tangent points right and the parabola bends upward, so $\mathbf N=\left\langle 0,1\right\rangle$.

Center and equation $$C=(0,0)+\frac 12\left\langle 0,1\right\rangle=\left(0,\frac 12\right),$$ $$\boxed{x^2+\left(y-\frac 12\right)^2=\frac 14}.$$

Geometry The center lies on the concave side of the curve, above the horizontal tangent. The circle has second-order contact with the parabola at the vertex.