SUMMARY
The discussion focuses on solving a homework problem involving the equation of a sphere. Part (a) correctly identifies the equation of the sphere passing through the point (6, -2, 3) with center (-1, 2, 1) as (x + 1)² + (y - 2)² + (z - 1)² = 69. In part (b), the intersection of the sphere with the yz-plane is determined by setting x = 0, leading to the equation 1 + (y - 2)² + (z - 1)² = 69. Part (c) confirms the center of the sphere given by the equation x² + y² + z² - 8x + 2y + 6z + 1 = 0 as (4, -1, -3) and the radius as 5.
PREREQUISITES
- Understanding of the equation of a sphere in three-dimensional space.
- Familiarity with coordinate geometry, specifically the yz-plane.
- Ability to manipulate algebraic equations to find intersections.
- Knowledge of completing the square in the context of sphere equations.
NEXT STEPS
- Study the derivation of the standard form of a sphere's equation.
- Learn about intersections of geometric shapes in three-dimensional space.
- Explore the concept of completing the square for quadratic equations.
- Review methods for finding centers and radii of spheres from general equations.
USEFUL FOR
Students studying geometry, particularly those focusing on three-dimensional shapes, as well as educators looking for examples of sphere equations and their applications in coordinate systems.