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Math 32A Quiz 8B Solution Name: SID: Problem 1. Find the two points on the ellipsoid x2 y 2 + + z2 = 1 4 9 where the tangent plane is normal to v = h1, 1, −2i. Solution: If F (x, y, z) is differentiable and (a, b, c) ∈ R3 is a point such that F (a, b, c) = 0, then ∇F (a, b, c) is normal to the tangent plane of the surface F (x, y, z) = 0 at (a, b, c). Thus, rewriting our ellipsoid equation as 0 = F (x, y, z) = x2 y 2 + + z 2 − 1, 4 9 we know that for any (a, b, c) on the surface, a vector is normal to the tangent a 2b plane at (a, b, c) if and only if it is parallel (i.e. a nonzero multiple of) ∇F (a, b, c) = 2 , 9 , 2c . Therefore, (a, b, c) has a tangent plane normal to v if and only if there is some non-zero real number λ such that 2b a = λ, = λ, and 2c = −2λ. 2 9 This gives us up to two independent equations; we will use 9 a = −2c, and b = − c. 2 Plugging back into the ellipsoid equation, we have 9 1 = c2 + c2 + c2 4 17 2 = c 4 4 =⇒ c2 = 17 2 =⇒ c = ± √ . 17 Plugging back into (1), this shows us that the solutions are the two vectors 4 9 2 −√ , −√ , √ and 17 17 17 4 9 2 √ , √ , −√ . 17 17 17 1 (1)