find S11 for x square+y square -4x-6y +11=0 at the point 2,4
Answers
Answer:#2 in 11.6: Find the equation of a tangent plane and the equation of
a normal line to the surface
x
2 + y
2 − z
2 = 18
at the point P(3, 5, −4).
Let f = x
2+y
2−z
2
. Then the surface is a level surface of f. Therefore,
the gradient of f at P is normal to the surface. We compute this vector:
∇f = h2x, 2y, −2zi; ∇f(P) = h6, 10, 8i.
The tangent plane at P has equation
6(x − 3) + 10(y − 5) + 8(z + 4) = 0.
The normal line at P is described by the parametric equations:
x = 3 + 6t, y = 5 + 10t, z = −4 + 8t.
• # 18 in 11.6: Find parametric equations for the line tangent to the
curve given by the intersection of the surfaces
x
2 + y
2 = 4 and x
2 + y
2 − z = 0
at the point P(
√
2,
√
2, 4).
The idea is to compute two normal vectors, and then compute their
cross product to produce a vector which is tangent to both surfaces
and, hence, tangent to their intersection.
Let f = x
2 + y
2
, and g = x
2 + y
2 − z. Compute their gradients and
evaulate at P:
∇f(P) = h2
√
2, 2
√
2, 0i, ∇g(P) = h2
√
2, 2
√
2, −1i.
The cross product of these two vectors is (±2
√
2)h1, −1, 0i Thus, the
following are parametric equations for the tangent line:
x =
√
2 + 2√
2t, y =
√
2 − 2
√
2t, z = 0
Step-by-step explanation: