Find the Center, Foci, and Vertices of the Ellipse $2 x ^ { 2 } + 5 y ^ { 2 } - 20 x + 30 y + 85 = 0$

question 5

Multiple Choice

Find the center, foci, and vertices of the ellipse.
- $2 x ^ { 2 } + 5 y ^ { 2 } - 20 x + 30 y + 85 = 0$

• A) \(\frac { ( x - 5 ) ^ { 2 } } { 2 } + \frac { ( y + 3 ) ^ { 2 } } { 5 } = 1\)
• center: \(( 5 , - 3 )\); foci: \(( 6.7 , - 3 ) , ( 3.3 , - 3 )\); vertices: \(( 7.2 , - 3 ) , ( 2.8 , - 3 )\)
• B) \(\frac { ( x - 5 ) ^ { 2 } } { 5 } + \frac { ( y + 3 ) ^ { 2 } } { 2 } = 1\)
• center: \(( - 5,3 )\); foci: \(( - 3.3,3 ) , ( - 6.7,3 )\); vertices: \(( - 7.2,3 ) , ( - 2.8,3 )\)
• C) \(\frac { ( x - 5 ) ^ { 2 } } { 2 } + \frac { ( y + 3 ) ^ { 2 } } { 5 } = 1\)
• center: \(( - 5,3 )\); foci: \(( - 3.3,3 ) , ( - 6.7,3 )\); vertices: \(( - 7.2,3 ) , ( - 2.8,3 )\)
• D) \(\frac { ( x - 5 ) ^ { 2 } } { 5 } + \frac { ( y + 3 ) ^ { 2 } } { 2 } = 1\)
• center: \(( 5 , - 3 )\); foci: \(( 6.7 , - 3 ) , ( 3.3 , - 3 )\); vertices: \(( 7.2 , - 3 ) , ( 2.8 , - 3 )\)

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Definitions:

Risk-Free Return

The theoretical return on an investment with zero risk, typically represented by government securities like Treasury bills.

Beta

A measure of a stock's volatility in relation to the overall market; a beta greater than 1 indicates higher than market volatility.

Treynor Measure

A performance metric for determining how well an investment compensates the investor for the risk taken, considering systematic risk.

Risk-Free Return

The theoretical return of an investment with zero risk, often represented by the yield of government bonds like U.S. Treasury bonds.