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

question 10

Multiple Choice

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
- $$\frac { ( y - 1 ) ^ { 2 } } { 4 } - \frac { ( x - 3 ) ^ { 2 } } { 49 } = 1$$

• A) center: \(( 3,1 )\)
• transverse axis is parallel to \(\mathrm { y }\)-axis
• vertices: \(( 3,1 - \sqrt { 53 } )\) and \(( 3,1 + \sqrt { 53 } )\);
• foci: \(( 3 , - 1 )\) and \(( 3,3 )\)
• asymptotes of \(y - 1 = - \frac { 7 } { 2 } ( x - 3 )\) and \(y - 1 = \frac { 7 } { 2 } ( x - 3 )\)
• B) center: \(( 3,1 )\)
• transverse axis is parallel to \(\mathrm { y }\)-axis
• vertices: \(( 1,0 )\) and \(( 4,4 )\)
• foci: \(( 1,2 - \sqrt { 53 } )\) and \(( 4,2 + \sqrt { 53 } )\)
• asymptotes of \(y - 1 = - \frac { 7 } { 2 } ( x - 3 )\) and \(y - 1 = \frac { 7 } { 2 } ( x - 3 )\)
• C) center: \(( 3,1 )\)
• transverse axis is parallel to \(y\)-axis
• vertices: \(( 3 , - 1 )\) and \(( 3,3 )\)
• foci: \(( 3,1 - \sqrt { 53 } )\) and \(( 3,1 + \sqrt { 53 } )\)
• asymptotes of \(\mathrm { y } - 1 = - \frac { 2 } { 7 } ( x - 3 )\) and \(y - 1 = \frac { 2 } { 7 } ( x - 3 )\)
• D) center: \(( - 3 , - 1 )\)
• transverse axis is parallel to \(x\)-axis
• vertices: \(( - 3 , - 3 )\) and \(( - 3,1 )\)
• foci: \(( - 3 , - 1 - \sqrt { 53 } )\) and \(( - 3 , - 1 + \sqrt { 53 } )\)
• asymptotes of \(y - 1 = - \frac { 2 } { 7 } ( x - 3 )\) and \(y - 1 = \frac { 2 } { 7 } ( x - 3 )\)

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Definitions:

Average Total Cost

The per-unit cost of production, obtained by dividing total costs by the total quantity of goods or services produced.

Marginal Cost

The expenditure for assembling another unit of a product or service.

Short-Run Equilibrium

A state in which market supply and demand balance each other, and as a result, prices become stable for a short period.

Constant Returns

A situation in economics where increasing the scale of production does not affect the long-run average cost of production, implying it remains constant.