Question

How to find the vertices of the hyperbole

Answer 1

The answer is the second one

If you want a fast explanation

You have to remember that the minus sign indicates which direction the hyperbole will follow, if the minus is on x, that indicates the hyperbole will be vertical, and if the minus is in y, then it'll be horizontal

If you check the vertices points those indicates the length of 2a thus thatll be 6

To get the center just use middle point equation and you'll get (1,3)

Just to know, a indicates the distance from the center to the vertices, b indicates how wide the hyperbole box is, and c indicates the distance from center to focis

A=3

B=?

C=6

We use Pythagoras so

$b = \sqrt{(6) ^{2} - (3) ^{2} }$

Thus you get

$b = \sqrt{27}$

With that data now you can get the equation

You know that below (y-3)^2 there should be a^2 so that means there will be the 9

And in the (x-1)^2 there should be b^2 so that means there will be the 27

PD. The 3 besides y, and 1 besides x represent the center

Answer 2

Answer:

(y - 3)²/9 - (x - 1)²/27 = 1

Step-by-step explanation:

1. Identify the centre of the hyperbola.

The centre is half-way between the two vertices.

((x₂ + x₁)/2, (y₂ + y₁)/2) = ((1 + 1)/2, (6 + 0)/2) = (2/2, 6/2) = (1, 3)

The centre is at (1, 3).

(1, 3) = (h, k), so

h = 1, k = 3

2. Identify the type of hyperbola.

Plot the vertices, foci, and centre.

We can see from Fig. 1 that we have a vertical hyperbola.

The equation for a vertical hyperbola is

(y - k)²/a² - (x - h)²/b² = 1

3. Identify the value of a

Count from the centre to either vertex.

a = 3

4. Identify the value of c.

Count from the centre to either focus.

c = 6

5. Identify the value of b

a² + b² =   c²

3² + b² =   6²

9 + b² = 36

Subtract 9 from each side

b² = 27

Take the square root of each side

b = √27

6. Write the equation for the hyperbola

We now have all the values needed: h, k, a, and b

The equation for the hyperbola is

(y - 3)²/9 - (x - 1)²/27 = 1

The graph of your hyperbola is shown in Fig. 2.