Find a vector rt that describes the curve C of intersection

Find a vector r(t) that describes the curve C of intersection between the given surfaces. Sketch the curve C. Use the indicated parameter.

x²+y²=9, z=9-x²; x=3cost

Solution

Given:x2+y2=9,

z=9-x2;

x=3cost;

if x=3cost then y^2=9-x^2

=9-(3cost)^2

=9-9cost^2

=9(1-cost^2)

=9sint^2

y=3sint

z=9-x^2

=9-(3cost)^2

=9-9cost^2

=9(1-cost^2)

=9sint^2

then  r(t)=xi+yj+zk

r(t)=3cost i+3sint j+9sint^2 k.