Answer:
option A : [tex]\frac{(y+5)^2}{9} - \frac{(x-2)^2}{3}=1[/tex]
Step-by-step explanation:
the equation of the hyperbola with center (2,-5), vertex (2,-2), and focus (2+-5+2sqrt3)
center is (2,-5), vertex is (2,-2). It is a vertical hyperbola
General equation for vertical hyperbola is
[tex]\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2}=1[/tex]
Center (2,-5) so h=2, k= -5
vertex is (2,-2)
We know vertex is (h, k+a), k=-5
k + a= -2
-5 + a = -2
so a = 3
Given focus (2+-5+2sqrt3)
Focus is (h , k+c), k= -5
[tex]k+c= -5+2\sqrt{3}[/tex]
[tex]-5+c= -5+2\sqrt{3}[/tex]
Add 5 on both sides
[tex]c= 2\sqrt{3}[/tex]
We need to find out b
c^2 = a^2 + b^2
[tex](2\sqrt{3})^2= 3^2 + b^2[/tex]
12 = 9 + b^2
b^2 = 3
we know a=3 so a^2 =9
we know h=2 and k = -5
Plug in all the values in general equation
[tex]\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2}=1[/tex]
[tex]\frac{(y+5)^2}{9} - \frac{(x-2)^2}{3}=1[/tex]
