 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/12/c/solution

A plane through the origin has equation

l x + m y + n z = 0

for some nonzero normal

(l, m, n)

. Dividing that normal by its Euclidean norm makes

l^{2} + m^{2} + n^{2} = 1

without changing the plane. Substitute the spherical coordinates from (a) and cancel the nonzero radius:

l sin θ cos ϕ + m sin θ sin ϕ + n cos θ = 0, l^{2} + m^{2} + n^{2} = 1 .

Its intersection with the sphere is the desired great circle; changing the sign of all three coefficients describes the same circle.

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