Verified answer

The intersection of the sphere centered at the origin and the plane z = 2 is a circle.

So C is a circle on the level of z = 2

x² + y² + z² = 26

With z = 2, we have

x² + y² + 4 = 26

x² + y² = 22

So now let x = √22cos(t) and y = √22sin(t) and let z = 2

Then the parametric curve is

r(t) = <√22cos(t), √22sin(t), 2> with 0≤ t≤ 2π

To find the tangent line to C we need to solve for a common parameter at the point (3,4,2) and find r'(t). So we need to solve the system of equations first:

√22cos(t) = 3

√22sin(t) = 4

which gives me

tan(t) = 4/3 ⇒ t = arctan(4/3)

Now find r'(t)

r'(t) = <-√22sin(t), √22cos(t), 0>

r'(arctan(4/3)) = <-4√22/5, 3√22/5, 0>

Then the tangent line at P(3,4,2) is

q(s) = <3, 4, 2> + s <-4√22/5, 3√22/5, 0>

Yin