 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ia/paper-3/3c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

For the parametrized curve

x (u) = (u, f (u))

κ = \frac{∣ x ^{'} y ^{''} - y ^{'} x ^{''} ∣}{( x ^{'2} + y ^{'2} ) ^{3/2}} = \frac{∣ f ^{''} ( u ) ∣}{( 1 + f ^{'} ( u ) ^{2} ) ^{3/2}} .

For the upper semicircle

f (u) = a^{2} - u^{2}

f^{'} = - \frac{u}{f}, f^{''} = - \frac{a ^{2}}{f ^{3}}, 1 + f^{'2} = \frac{a ^{2}}{f ^{2}},

and therefore $κ = 1/ a$ , as required for a circle of radius

a

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