Complete the Square to Determine Whether the Equation $y ^ { 2 } + 25 = 14 y - 16 x$

question 96

Essay

Complete the square to determine whether the equation $y ^ { 2 } + 25 = 14 y - 16 x$ represents an ellipse, parabola, hyperbola or 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 axis. 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:

Cactus Spines

Modified leaves or parts of leaves that have evolved into sharp structures to help reduce water loss and deter herbivores.

Modified

Changed or altered from its original form or function, often through natural processes, human intervention, or technological means.

Trichomes

Small hair-like outgrowths from the epidermis of plants, serving various functions such as reducing water loss and providing protection against predators.

Specialized Stems

Stems that perform functions other than support and conduction, including storage, reproduction, and photosynthesis, such as rhizomes, tubers, and cacti.