Graph the Hyperbola $3 y ^ { 2 } - 7 x ^ { 2 } = 1$

question 3

Multiple Choice

Graph the hyperbola. Specify the following: vertices, foci, lengths of transverse and conjugate axes, eccentricity, and equations of the asymptotes. $3 y ^ { 2 } - 7 x ^ { 2 } = 1$

• A) vertices: \(\left( \pm \frac { \sqrt { 7 } } { 7 } , 0 \right)\) ; Foci: \(\left( \pm \frac { \sqrt { 210 } } { 21 } , 0 \right)\) ; Length of transverse axis: \(\frac { 2 \sqrt { 3 } } { 3 }\) ; Length of conjugate axis: \(\frac { 2 \sqrt { 7 } } { 7 }\) ; Eccentricity: \(\frac { \sqrt { 70 } } { 7 }\) ; Asymptotes: \(y = \pm \frac { \sqrt { 21 } } { 3 } x\) .
• B) vertices: \(\left( \pm \frac { \sqrt { 3 } } { 6 } , 0 \right)\) ; Foci: \(\left( \pm \frac { 2 \sqrt { 3 } } { 3 } , 0 \right)\) ; Length of transverse axis: \(\frac { \sqrt { 3 } } { 3 }\) ; Length of conjugate axis: \(2\) ; Eccentricity: \(2\) ; Asymptotes: \(y = \pm \frac { \sqrt { 3 } } { 3 } x\) .
• C) vertices: \(\left( 0 , \pm \frac { \sqrt { 3 } } { 3 } \right)\) ; Foci: \(\left( 0 , \pm \frac { \sqrt { 210 } } { 21 } \right)\) ; Length of transverse axis: \(\frac { 2 \sqrt { 3 } } { 3 }\) ; Length of conjugate axis: \(\frac { 2 \sqrt { 7 } } { 7 }\) ; Eccentricity: \(\frac { \sqrt { 70 } } { 7 }\) ; Asymptotes: \(y = \pm \frac { \sqrt { 21 } } { 7 } x\) .
• D) vertices: \(\left( \pm \frac { \sqrt { 7 } } { 7 } , 0 \right)\) ; Foci: \(\left( \pm \frac { \sqrt { 210 } } { 21 } , 0 \right)\) ; Length of transverse axis: \(\frac { 2 \sqrt { 7 } } { 7 }\) ; Length of conjugate axis: \(\frac { 2 \sqrt { 3 } } { 3 }\) ; Eccentricity: \(\frac { \sqrt { 70 } } { 7 }\) ; Asymptotes: \(y = \pm \frac { \sqrt { 21 } } { 7 } x\) .
• E) vertices: \(\left( 0 , \pm \frac { \sqrt { 3 } } { 3 } \right)\) ; Foci: \(\left( 0 , \pm \frac { \sqrt { 210 } } { 21 } \right)\) ; length of transverse axis: \(\frac { 2 \sqrt { 3 } } { 3 }\) ; Length of conjugate axis: \(\frac { 2 \sqrt { 7 } } { 7 }\) ; Eccentricity: \(\frac { \sqrt { 70 } } { 7 }\) ; Asymptotes: \(y = \pm \frac { \sqrt { 21 } } { 3 } x\) .

Compute deadweight loss and the coefficient of monopsony power in monopsonistic markets.

Differentiate between monopsony, monopoly, bilateral monopoly, and oligopsony market structures.

Identify the effects of changes in market conditions (e.g., supply and demand elasticity, market supply shifts) on monopsony outcomes.

Examine regulatory effects and antitrust laws on monopolies and their impact on pricing and competition.

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Definitions:

Indifference Curve

A graphical representation showing different bundles of goods between which a consumer is indifferent, reflecting their preferences and utility.

Total Utility

The overall satisfaction or benefit a consumer receives from consuming a particular quantity of a good or service.

Indifference Curve

A graphical representation showing combinations of goods or services among which a consumer is indifferent, meaning they have no preference for one combination over another, holding utility constant.

Absolute Value

The non-negative value of a number without regard to its sign.