Complete the Square to Determine Whether the Equation $y ^ { 2 } + 2 y - 12 x + 38 = 0$

question 47

Essay

Complete the square to determine whether the equation $y ^ { 2 } + 2 y - 12 x + 38 = 0$ 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.

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Definitions:

Mitral Valve

A valve in the heart that lies between the left atrium and left ventricle, ensuring unidirectional blood flow through the heart.

Chest Tube

is a medical device inserted into the pleural space to remove air, fluids, or pus, often used in treating conditions like pneumothorax or pleural effusion.

Bubbling

The production of bubbles in a liquid caused by processes such as boiling, fermentation, or chemical reactions.

Water-Seal Chamber

A part of a chest drain system that prevents air from re-entering the chest cavity while allowing air and fluids to escape from it.