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)