Complete the Square to Determine Whether the Equation $x ^ { 2 } + 4 x - 7 y ^ { 2 } + 70 y = 171$

question 57

Essay

Complete the square to determine whether the equation $x ^ { 2 } + 4 x - 7 y ^ { 2 } + 70 y = 171$ , 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.

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Definitions:

Transgenic Crops

Crops that have been genetically modified to express a gene from another species, conferring beneficial traits such as pest resistance or increased yield.

Pesticide-Resistant

Refers to pests that have developed a resistance to pesticides through genetic mutation or adaptation, reducing the effectiveness of the pesticides.

Pest-Resistant

A characteristic of plants that have the ability to withstand or repel attacks from pests, reducing the need for chemical pesticides.

Gene Flow

The movement of alleles between local populations due to the migration of individuals; can have significant evolutionary consequences.