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Question: 12 given a ruled surface m parametrized by xu ...

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12. Given a ruled surface M parametrized by x(u, υ-α (u) + υβ(u) with α, 0 and lli 1. Check that we may assume that α(u) . β(u) I (u)β(u) for a suitable function t.) Suppose, moreover, that α(u), β (u), and β(u) are linearly dependent for every u. conclude that β(u) A(11)a,(u) for some function λ. Prove that: (i) IfX(u)-0 for all u, then M is a cylinder (ii) If λ is a nonzero constant, then M is a cone. 0 for all u. (Hint: Replace α(u) with α(u) + a. b. (iii) Ifλ and λ, are both nowhere zero, then M is a tangent developable. (Hint: Find the directrix )
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