Write the Equation of the Ellipse in Standard Form $$4 x ^ { 2 } + 9 y ^ { 2 } + 12 x - 108 y + 297 = 0$$

question 33

Multiple Choice

Write the equation of the ellipse in standard form. Identify the center and foci.
- $$4 x ^ { 2 } + 9 y ^ { 2 } + 12 x - 108 y + 297 = 0$$

• A) \(\frac { \left( x + \frac { 3 } { 2 } \right) ^ { 2 } } { 9 } + \frac { ( y - 6 ) ^ { 2 } } { 4 } = 1\)
• center: \(\left( - \frac { 3 } { 2 } , 6 \right)\)
• foci: \(\left( - \frac { 3 } { 2 } - \sqrt { 5 } , 6 \right) , \left( - \frac { 3 } { 2 } + \sqrt { 5 } , 6 \right)\)
• B) \(\frac { \left( x + \frac { 3 } { 2 } \right) ^ { 2 } } { 9 } + \frac { ( y - 6 ) ^ { 2 } } { 4 } = 1\)
• center: \(\left( - \frac { 3 } { 2 } , 6 \right)\)
• foci: \(\left( - \frac { 9 } { 2 } , 6 \right) , \left( \frac { 3 } { 2 } , 6 \right)\)
• C) \(\frac { \left( x + \frac { 3 } { 2 } \right) ^ { 2 } } { 9 } + \frac { ( y - 6 ) ^ { 2 } } { 4 } = 1\)
• center: \(\left( \frac { 3 } { 2 } , - 6 \right)\)
• foci: \(\left( \frac { 3 } { 2 } - \sqrt { 5 } , - 6 \right) , \left( \frac { 3 } { 2 } + \sqrt { 5 } , - 6 \right)\)
• foci: \(\left( \frac { 3 } { 2 } - \sqrt { 5 } , - 6 \right) , \left( \frac { 3 } { 2 } + \sqrt { 5 } , - 6 \right)\)
• D) \(\frac { \left( x - \frac { 3 } { 2 } \right) ^ { 2 } } { 9 } + \frac { ( y + 6 ) ^ { 2 } } { 4 } = 1\)
• center: \(\left( \frac { 3 } { 2 } , - 6 \right)\)
• foci: \(\left( \frac { 3 } { 2 } - \sqrt { 5 } , - 6 \right) , \left( \frac { 3 } { 2 } + \sqrt { 5 } , - 6 \right)\)

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Definitions:

≠ Symbol

A mathematical symbol representing inequality, indicating that two values are not equal.

Alternative Hypothesis

A hypothesis that contradicts the null hypothesis, suggesting a new effect or relationship.

Directional

Refers to tests or hypotheses that specify the expected direction of the relationship between variables (e.g., greater than, less than).

Directional

In research, directional usually refers to a hypothesis that specifies the direction of an expected difference or relationship.