 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-2/6/d/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The center of the sphere-plane intersection is the perpendicular projection of the sphere's center onto the plane:

c_{\circ} = c + \frac{20 - n \cdot c}{∣ n ∣ ^{2}} n = (1, 2, 3) + (1, 2, 2) = (2, 4, 5) .

The distance from

c

to the plane is

3 < 4

, so the intersection is a genuine circle. A right triangle with hypotenuse equal to the sphere radius gives

ρ = 4^{2} - 3^{2} = 7 .

Its plane is the given plane; displacements within it are perpendicular to

n

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