Analyzing a Hyperbola: Finding Center, Vertices, Foci, and Asymptotes

In this article, we will explore the process of finding the center, vertices, foci, and asymptotes of a hyperbola given by the equation 4x^2 - y^2 - 8x - 8y - 112 0. We will go through the steps to rewrite the equation in standard form and then identify each component.

Step 1: Rearrange the Equation

Let's start by rearranging the equation:

4x^2 - y^2 - 8x - 8y - 112 0

Step 2: Group and Complete the Square

We will group and complete the square for both x and y.

For x

Take 4x^2 - 8x. Factor out the 4: 4(x^2 - 2x). Complete the square inside the parentheses: x^2 - 2x (x - 1)^2 - 1. So, 4(x^2 - 2x) 4((x - 1)^2 - 1) 4(x - 1)^2 - 4.

For y

Take -y^2 - 8y. Factor out -1: -1(y^2 8y). Complete the square inside the parentheses: y^2 8y (y 4)^2 - 16.

Step 3: Substitute Back into the Equation

Now substitute back into the equation:

4(x - 1)^2 - 4 - (y 4)^2 16 - 112 0

Simplifying gives:

4(x - 1)^2 - (y 4)^2 - 100 0

Rearranging gives:

4(x - 1)^2 - (y 4)^2 100

Step 4: Divide by 100 to Get Standard Form

Divide each term by 100:

(frac{(x - 1)^2}{25} - frac{(y 4)^2}{100} 1)

Step 5: Identify Components

This is now in the standard form of a hyperbola:

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

where:

k - 4 a^2 25 (Rightarrow a 5) b^2 100 (Rightarrow b 10)

Center of the Hyperbola

The center of the hyperbola is:

(-1, -4)

Vertices

The vertices are located a units from the center along the x-axis:

Right vertex: (-1 5, -4) (4, -4) Left vertex: (-1 - 5, -4) (-6, -4)

Foci

The distance to the foci is given by c, where c sqrt{a^2 b^2}

c sqrt{25 100} sqrt{125} 5sqrt{5}

The foci are located c units from the center along the x-axis:

Right focus: (-1 5sqrt{5}, -4) Left focus: (-1 - 5sqrt{5}, -4)

Asymptotes

The equations of the asymptotes for a hyperbola of this form are given by:

y - k pm frac{b}{a}(x - h)

Substituting:

h -1 k -4 a 5 b 10

gives:

y - (-4) pm frac{10}{5}(x - (-1))

Simplifying gives:

y 4 pm 2(x 1)

Thus the equations of the asymptotes are:

y 2x 6 y -2x - 2

Summary of Results

Center: (-1, -4) Vertices: (4, -4) and (-6, -4) Foci: (-1 5sqrt{5}, -4) and (-1 - 5sqrt{5}, -4) Asymptotes: y 2x 6 and y -2x - 2