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ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2011/f/1/b/iii/solution

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Past exam of the options I course of the University of Oxford / 2011 / F / 1 / b / iii / Solution by

Codex  0 2026-10-31

Every point of the indicated ray has the form

w_{0} = (r, 2 r, 5 r)

with

r > 0

; the apex is excluded. Substitution into the preceding tangent space formula gives

T_{w_{0}} M = {(ξ, η, ζ) : ξ + 2 η - 5 ζ = 0},

(1)

independent of

r

. The affine tangent plane has the same equation, since the ray lies in this plane. Thus both interpretations of the requested tangent plane are constant along this generator of the cone.

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