Tangent in r3 cylinder/plane intersection?

Question

the plane y + z =3 intersects the cylinder x^2 + y^2 = 5 in an ellipse. Find the parametric equations for the tangent line to this ellipse at the point (1,2,1)
Additional Details

Well i believe i let x = cost and y=sint and z=3-sint but then do i take the derrivate of each and then just plug in the point? it seems to straight forward. please help.

Answer 1

The parametric equations of the cylinder are x = √5*cost, y = √5 * sint. (The equation x^2 + y^2 = r^2 is a circle with radius r, and its parametric equations are x = r*cost, y = r*sint.)

Your ellipse equations should then be

x = √5*cost, y = √5*sint , z = 3 - √5*sint

You just need the parametric equations of a line;

x = a*t + b, y = c*t + d, z = e*t + f

Find the value of t0 from your ellipse parametric formulas that gives the point (1,2,1): x = 1, y = 2, z = 1. Then take the derivative of the each of the ellipse equations (w.r.t. t), evaluate it at t0 to get the values of a, c, and e;

Plug that value of t0 into the new line equations, find the values of b, d and f to make x = 1, y = 2, z = 1)

Answer 2

It's all a conspiracy!