Find the Center, Transverse Axis, Vertices, Foci, and Asymptotes of the Hyperbola

question 36

Multiple Choice

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
- $$( x - 1 ) ^ { 2 } - 4 ( y - 2 ) ^ { 2 } = 4$$

• A) center at \(( 1,2 )\)
• transverse axis is parallel to \(\mathrm { y }\)-axis
• vertices at \(( 1,0 )\) and \(( 1,4 )\),
• foci at \(( 1,2 - \sqrt { 5 } )\) and \(( 1,2 + \sqrt { 5 } )\),
• asymptotes of \(y + 2 = - 2 ( x + 1 )\) and \(y + 2 = 2 ( x + 1 )\)
• B) center at \(( 1,2 )\)
• transverse axis is parallel to \(x\)-axis
• vertices at \(( 0,2 )\) and \(( 2,2 )\)
• foci at \(( 1 - \sqrt { 5 } , 2 )\) and \(( 1 + \sqrt { 5 } , 2 )\)
• asymptotes of \(y - 2 = - 2 ( x - 1 )\) and \(y - 2 = 2 ( x - 1 )\)
• C) center at \(( 1,2 )\)
• transverse axis is parallel to \(x\)-axis
• vertices at \(( - 1,2 )\) and \(( 3,2 )\)
• foci at \(( 1 - \sqrt { 5 } , 2 )\) and \(( 1 + \sqrt { 5 } , 2 )\)
• asymptotes of \(y - 2 = - \frac { 1 } { 2 } ( x - 1 )\) and \(y - 2 = \frac { 1 } { 2 } ( x - 1 )\)
• D) center at \(( 2,1 )\)
• transverse axis is parallel to \(x\)-axis
• vertices at \(( 0,1 )\) and \(( 4,1 )\)
• foci at \(( 2 - \sqrt { 5 } , 1 )\) and \(( 2 + \sqrt { 5 } , 1 )\)
• asymptotes of \(y - 1 = - \frac { 1 } { 2 } ( x - 2 )\) and \(y - 1 = \frac { 1 } { 2 } ( x - 2 )\)

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Definitions:

Theoretical Minimum

The lowest possible amount of resources, energy, or time required to achieve a particular process or operation, often used as a benchmark in process improvement.

Network Diagram

A visual representation of a project's activities and their sequence, showing the path from start to finish, including dependencies.

Cycle Time

The total time from the beginning to the end of a process, which includes both processing and waiting times.

Maximum Cycle Time

The longest duration permitted for completing one cycle of a production process, defining the upper limit of process efficiency.