 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ia/paper-1/5c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Write

x = a + u, y = a - u .

The first equation is then automatic, while the second becomes

x \cdot y = a \cdot a - u \cdot u = c .

x = a + u, y = a - u, ∣ u ∣ = a \cdot a - c .

u

varies,

x

and

y

are antipodal points of the sphere with center

a

a \cdot a - c

, so they are opposite ends of a diameter.

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