Final answer:
To reduce the equation to standard form, complete the square for x and z terms to reveal the equation represents a degenerate elliptic cone which is reduced to a single point.
Step-by-step explanation:
The student asked to reduce the equation x² + 4y² + 4z² + 4x - 16z + 16 = 0 to one of the standard forms.
First, we need to group the x terms and z terms and look for opportunities to complete the square for both x and z:
- x² + 4x can be completed by adding and subtracting (4/2)² = 4
- 4z² - 16z can be completed by adding and subtracting (16/2²) = 16
So, the equation becomes:
x² + 4x + 4 + 4(z² - 4z + 4) + 4y² - 4 - 16 + 16 = 0
Which simplifies to:
(x + 2)² + 4(y² + (z - 2)²) = 0
This is close to the standard form of an elliptic cone, except all terms are on one side and equal zero, which indicates the cone degenerates into a single point.