Final answer:
A parametrization of the curve formed at the intersection of the given surfaces can be found using trigonometric functions or by directly enforcing a relationship between y and z. Two valid parametrizations include using a circular relationship for y and z and solving for x, or assigning y to t and deriving z and x accordingly.
Step-by-step explanation:
The student's question requests a parametrization of the curve of intersection of two surfaces, y²−z²=x−2 and y²+z²=9 with z≥0. To find a parametrization, we can set y to some function of t, and derive z and x from the given equations. One of the ways to parametrize this is by using a trigonometric function for y since y²+z²=9 is a circle of radius 3 in the yz-plane. We can let y=3cos(t) and z=3sin(t) with t in the interval [0, π] because z is nonnegative. Then from y²−z²=x−2, we have (3cos(t))²−(3sin(t))²=x−2, simplifying to 9cos(2t)=x−2, thus x=9cos(2t)+2.
Second parametrization can be created by considering the y²−z²=x−2 equation. Setting y=t, we can solve for z using y²+z²=9, which gives us z=√(9−t²). Thus, z is nonnegative when 0≤t≤3. The value of x can then be found using y²−z²=x−2, which gives x=t²-√(9−t²)²+2.