Find the standard form of the equation of the hyperbola with the given characteristics. foci: asymptotes:

Question

Find the standard form of the equation of the hyperbola with the given characteristics. foci:  asymptotes:

Quizplus · Accepted Answer

The given foci and asymptotes give the following information:
- The distance between the foci is 2ae = 11, where a is the distance from the center to a vertex and e is the eccentricity.
- The slopes of the asymptotes are ±b/a, where b is the distance from the center to the intersection of the hyperbola with its transverse axis.
Using the fact that the transverse axis is perpendicular to the asymptotes, we can find that b/a = ±3/4. Squaring both sides gives b^2/a^2 = 9/16.
Now we have two equations with two unknowns (a and e):
2ae = 11
b^2/a^2 + e^2 = 1
Substituting b^2/a^2 = 9/16 into the second equation and solving for e gives e = √(1 - 9/16) = √(7/16) = √7/4.
Substituting this into the first equation and solving for a gives a = 11/(2e) = 11/(2√7/4) = 11√7/2.
Now we can use the standard form of the hyperbola centered at the origin:
x^2/a^2 - y^2/b^2 = 1
Substituting in the values we found gives:
x^2/(11√7/2)^2 - y^2/b^2 = 1
Simplifying the denominator and comparing to the choices given, we see that the answer is D).

Find the Standard Form of the Equation of the Hyperbola