The equation of the hyperbolas is [tex]\frac{x^2}{117} -\frac{y^2}{52} =1[/tex]
Hyperbolas are symmetrical open curve that are formed when a circular cone intersects a plane.
The given parameters are:
The equation of a hyperbola is represented as:
[tex]\frac{(x - h)^2}{a^2} -\frac{(y - k)^2}{b^2} =1[/tex]
The hyperbola passes through the origin.
So, we have:
[tex]\frac{(x - 0)^2}{a^2} -\frac{(y - 0)^2}{b^2} =1[/tex]
[tex]\frac{x^2}{a^2} -\frac{y^2}{b^2} =1[/tex]
Also, from the asymptotes, we have:
[tex]\frac ba = \frac 23[/tex]
Make b the subject
[tex]b= \frac 23a[/tex]
From the foci, we have:
[tex]a^2 + b^2 = 13^2[/tex]
Substitute [tex]b= \frac 23a[/tex]
[tex]a^2 + (\frac 23a)^2 = 13^2[/tex]
Evaluate the squares
[tex]a^2 + \frac 49a^2 = 13^2[/tex]
Add the fractions
[tex]\frac {13}9a^2 = 13^2[/tex]
Divide both sides by 13
[tex]\frac {1}9a^2 = 13[/tex]
Multiply both sides by 9
[tex]a^2 = 117[/tex]
Recall that:
[tex]b= \frac 23a[/tex]
Square both sides
[tex]b^2 = \frac 49a^2[/tex]
So, we have:
[tex]b^2 = \frac 49 \times 117[/tex]
[tex]b^2 = 52[/tex]
Recall that:
[tex]\frac{x^2}{a^2} -\frac{y^2}{b^2} =1[/tex]
So, we have:
[tex]\frac{x^2}{117} -\frac{y^2}{52} =1[/tex]
Hence, the equation of the hyperbolas is [tex]\frac{x^2}{117} -\frac{y^2}{52} =1[/tex]
Read more about hyperbolas at:
https://brainly.com/question/12919612
