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

question 135

Multiple Choice

Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
- $$x ^ { 2 } - 16 y ^ { 2 } + 6 x - 32 y - 23 = 0$$

• A) center at \(( - 3 , - 1 )\)
• transverse axis is parallel to \(x\)-axis
• vertices at \(( - 7 , - 1 )\) and \(( 1 , - 1 )\)
• foci at \(( - 3 - \sqrt { 17 } , - 1 )\) and \(( - 3 + \sqrt { 17 } , - 1 )\)
• asymptotes of \(y + 1 = - \frac { 1 } { 4 } ( x + 3 )\) and \(y + 1 = \frac { 1 } { 4 } ( x + 3 )\)
• B) center at \(( - 3 , - 1 )\)
• transverse axis is parallel to \(\mathrm { y }\)-axis
• vertices at \(( - 3 , - 5 )\) and \(( - 3,3 )\)
• foci at \(( - 3 , - 1 - \sqrt { 17 } )\) and \(( - 3 , - 1 + \sqrt { 17 } )\)
• asymptotes of \(\mathrm { y } - 1 = - 4 ( \mathrm { x } - 3 )\) and \(\mathrm { y } - 1 = 4 ( \mathrm { x } - 3 )\)
• C) center at \(( - 3 , - 1 )\)
• transverse axis is parallel to \(x\)-axis
• vertices at \(( - 4 , - 1 )\) and \(( - 2 , - 1 )\)
• foci at \(( - 3 - \sqrt { 17 } , - 1 )\) and \(( - 3 + \sqrt { 17 } , - 1 )\)
• asymptotes of \(\mathrm { y } + 1 = - 4 ( \mathrm { x } + 3 )\) and \(\mathrm { y } + 1 = 4 ( \mathrm { x } + 3 )\)
• D) center at \(( - 1 , - 3 )\)
• transverse axis is parallel to \(x\)-axis
• vertices at \(( - 5 , - 3 )\) and \(( 3 , - 3 )\)
• foci at \(( - 1 - \sqrt { 17 } , - 3 )\) and \(( - 1 + \sqrt { 17 } , - 3 )\)
• asymptotes of \(y + 3 = - \frac { 1 } { 4 } ( x + 1 )\) and \(y + 3 = \frac { 1 } { 4 } ( x + 1 )\)

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Definitions:

Long Run

An interval in which every aspect of production and costs can vary, enabling modifications to all contributions.

Supply

The whole sum of a service or product that is up for sale at a certain price tier in a specific marketplace.

Decreasing-Cost Industry

An industry where the average cost of production decreases as the industry's output increases, often due to economies of scale.

Long-Run Supply Curve

A graphical representation showing the relationship between the market price of a good and the quantity of it that producers are willing to supply when all production inputs are variable.