AnswersPrecalculusEquations of Hyperbolas — QuizQuestion 6
6
QuizMultiple Choice

Equations of Hyperbolas — Quiz

Question 6 • Precalculus

Which is a true statement comparing the graphs of the fraction with numerator x squared and denominator 6 squared minus the fraction with numerator y squared and denominator 8 squared is equal to 1 and the fraction with numerator x squared and denominator 8 squared minus the fraction with numerator y squared and denominator 6 squared is equal to 1 ?

Answer
A
The lengths of both transverse axes are the same.
B
The foci of both graphs are the same points.
C
The directrices of the fraction with numerator x squared and denominator 6 squared minus the fraction with numerator y squared and denominator 8 squared is equal to 1 are horizontal while the directrices of the fraction with numerator x squared and denominator 8 squared minus the fraction with numerator y squared and denominator 6 squared is equal to 1 are vertical.
D
The vertices of the fraction with numerator x squared and denominator 6 squared minus the fraction with numerator y squared and denominator 8 squared is equal to 1 are on the y -axis while the vertices of the fraction with numerator x squared and denominator 8 squared minus the fraction with numerator y squared and denominator 6 squared is equal to 1 are on the x -axis.

Explanation

For a horizontal hyperbola, the foci are at (±c, 0), where c² = a² + b². In the first equation, c² = 6² + 8² = 100, so c = 10; in the second, c² = 8² + 6² = 100 as well. Thus both graphs have foci at (−10, 0) and (10, 0), making “The foci of both graphs are the same points” true.

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