Final answer:
The equation represents an ellipse with a center at (0, 0), a major axis along the x-axis, and a minor axis along the y-axis. The distances between the center and the vertices along the major and minor axes are 8 and approximately 4.62, respectively.
Step-by-step explanation:
The equation x² - 2xy + 4y² = 64 represents a quadratic equation in two variables, x and y. This equation can be simplified by completing the square to obtain (x - y)² + 3y² = 64. From this equation, we can understand the following information:
- The graph of this equation will be a conic section, specifically an ellipse, since the coefficients of x² and y² are positive.
- The center of the ellipse is at the point (0, 0) since the equation contains only even powers of x and y.
- The major axis of the ellipse is along the x-axis, and the minor axis is along the y-axis since the coefficient of x² is larger than the coefficient of y².
- The distance between the center of the ellipse and the vertex along the major axis is equal to the square root of 64, which is 8.
- The distance between the center of the ellipse and the vertex along the minor axis is equal to the square root of 64/3, which is approximately 4.62.