Question

Write the equation of the hyperbola with center (2,-5), vertex (2,-2), and focus (2+-5+2sqrt3)

Answer 1

Answer:

option A : $\frac{(y+5)^2}{9} - \frac{(x-2)^2}{3}=1$

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

$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2}=1$

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

$k+c= -5+2\sqrt{3}$

$-5+c= -5+2\sqrt{3}$

Add 5 on both sides

$c= 2\sqrt{3}$

We need to find out b

c^2 = a^2 + b^2

$(2\sqrt{3})^2= 3^2 + b^2$

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

$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2}=1$

$\frac{(y+5)^2}{9} - \frac{(x-2)^2}{3}=1$

Answer 2

Answer:

Its A for sure :)