Discussion
The radius of curvature of \( y = x^{2} \) at the origin is
Answer: C
At the origin \( y' = 2x = 0 \) and \( y'' = 2 \), so the formula collapses to the reciprocal of the second derivative:
\[ \rho \;=\; \frac{(1 + 0)^{3/2}}{|2|} \;=\; \frac{1}{2}. \]
The parabola is therefore locally the circle \( x^{2} + (y - 1/2)^{2} = 1/4 \), whose centre sits on the axis at the height of the focus
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