Complete the Square to Determine Whether the Equation $64 x ^ { 2 } + y ^ { 2 } + 128 x - 10 y + 73 = 0$

question 89

Essay

Complete the square to determine whether the equation $64 x ^ { 2 } + y ^ { 2 } + 128 x - 10 y + 73 = 0$ represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.

---

Definitions:

Utility Function

A mathematical representation used in economics to describe how the quantities of goods or services consumed lead to a level of consumer satisfaction or utility.

Whisky Consumption

Whisky Consumption refers to the amount of whisky that is consumed over a certain period, often considered in economic studies of consumer behavior and market demand.

Gallons

A unit of volume used primarily in the United States to measure liquids.

Whisky Consumption

The act of drinking whisky as a means of enjoyment or recreation.