Question

Let c be the curve of intersection of the cylinder x² + y² = 2 and the hyperbolic paraboloid x² - y² z = 5. What is the equation of the curve c?

Answer 1

Upon solving the given equations of the cylinder and the hyperbolic paraboloid simultaneously, it has been determined that there is no real intersection between the cylinder x² + y² = 2 and the hyperbolic paraboloid x² - y² = 5.

Step-by-step explanation:

To find the equation of the curve c, which is the intersection of the cylinder x² + y² = 2 and the hyperbolic paraboloid x² - y² = z, we set z = 5. Substituting z = 5 into the hyperbolic paraboloid equation gives x² - y² = 5. Solving these two equations together:

• Equation of the cylinder: x² + y² = 2

• Equation of the hyperbolic paraboloid with z=5: x² - y² = 5

By solving these equations simultaneously, we find the equation representing curve c that satisfies both surfaces:

1. Subtract the second equation from the first to eliminate x²: 2y² = -3, which is impossible for real numbers and indicates there is no real intersection.