Question

Let C be the intersection of x^2+y^2=16 and x+y+z=5. Find the curvature at (0,4,1).

I don't know how to find the intersection between the given equations.

Answer 1

The parametric equations for the circle are

x = 4cos t
y = 4sin t

so, the curve of the intersection also includes

z = 5-x-y = 5-4cost-4sint

Now, the curvature of the curve described is

k = |dT/ds| = |dT/dt| ÷ |dr/dt|
at (0,4,1), t = π/2

r = <4cost,4sint,5-4cost-4sint>
r' = <-4sint,4cost,4sint-4cost>
at t = π/2, r' = <-4,0,4>
|r'| = 4√2

T = r'/|r| = 1/√2 <-4sint,4cost,4sint-4cost>
dT/dt = 1/√2 <-4cost,-4sint,4cost+4sint>
so, dT/dt = 1/√2 <0,-4,4>
|dT/dt| = 4

so, k = (4)/(4√2) = 1/√2

As always, double-check my math.