Problem 43 - Algebra and Trigonometry

Question

convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$ x^{2}-y^{2}-2 x-4 y-4=0 $$

Step-by-Step Solution

Answer

The standard form of the hyperbola is \((x-1)^2 - (y-2)^2 = 1\). The foci are located at (1 + √2, 2) and (1 - √2, 2). The equations of the asymptotes are y = x + 1 and y = -x + 3.

Step 1: Complete the Square

Rearrange and group the terms with x and y separately to prepare for completing the square. We'll add 4 to both sides: \[x^2 - 2x + y^2 - 4y = 4.\] Now, add the square of half of the coefficient of x, which is 1 to both sides: \[(x^2 - 2x + 1) - (y^2 - 4y + 4) = 4 + 1 - 4,\] which simplifies to \[(x-1)^2 - (y-2)^2 = 1.\]

Step 2: Identify a, b and the standard form

The standard form for a hyperbola with horizontal transverse axis is: \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\). By comparing, we find that a and b are both equal to 1 and the center of the hyperbola is at (h, k) or (1, 2).

Step 3: Find the Foci

The foci c is given by the equation \(c = \sqrt{a^2 + b^2}\). Substitute a = 1, b = 1 into the formula to get \(c = \sqrt{2}\). The foci will be located along the transverse axis, c units from the center in both directions which gives us: (1 + √2, 2) and (1 - √2, 2)

Step 4: Find the Asymptotes

The equations for the asymptotes of a horizontal hyperbola are given by \(y -k = \pm (b/a)(x - h)\). Substitute (h, k) = (1, 2), a = 1, b = 1 to get the equations of the asymptotes: y = x + 1 and y = -x + 3

Key Concepts

Completing the Square

Completing the square is a method used to rewrite a quadratic equation so that it reveals the vertex form of a conic section.
This technique is particularly useful when dealing with hyperbolas, as it helps in converting the given equation into its standard form.
Let's focus on how to apply completing the square in the provided exercise.

First, we need to rearrange our equation \( x^2 - y^2 - 2x - 4y - 4 = 0 \) by grouping the \( x \) and \( y \) terms.
This gives us: \( x^2 - 2x + y^2 - 4y = 4 \). We then complete the square for each variable:

• For \( x \): Take half of the coefficient of \( x \), square it, and add it to both sides.
  Half of \(-2\) is \(-1\), and its square is \(1\). Thus, we add 1 to both sides.
• For \( y \): Similarly, half of \(-4\) is \(-2\), and its square is \(4\). Add this to both sides too.

After performing these operations, the equation becomes:
\((x-1)^2 - (y-2)^2 = 1\), transforming the original expression into a more manageable form.

Standard Form of a Hyperbola

The standard form of a hyperbola equation reveals important features of its graph, including its orientation, center, and relative dimensions.
Once we've completed the square, the standard form for a horizontal hyperbola is:\[(x-h)^2/a^2 - (y-k)^2/b^2 = 1\]Here, \((h, k)\) is the center of the hyperbola, while \(a\) and \(b\) represent the distances from the center to the vertices along the transverse and conjugate axes respectively.

The exercise transformed into:\((x-1)^2 - (y-2)^2 = 1\), indicating:

• The center \((h, k)\) of the hyperbola at (1, 2).
• Both \(a\) and \(b\) equal 1, pointing to a symmetric transverse axis.

Understanding the standard form is crucial, as it not only aids in graphing the hyperbola but also in identifying other key features like the foci and asymptotes.

Foci of Hyperbola

Foci are specific points located along the transverse axis of a hyperbola, playing a vital role in its geometric definition.
In any hyperbola, the foci are determined using the equation:\[c = \sqrt{a^2 + b^2}\]Once \(a\) and \(b\) are known, \(c\) can be calculated to find the exact positions of the foci.

In our exercise, given \(a = 1\) and \(b = 1\), we have:\[c = \sqrt{1^2 + 1^2} = \sqrt{2}\].The foci are then located at:

• (1 + \(\sqrt{2}\), 2)
• (1 - \(\sqrt{2}\), 2)

These points are symmetrical and lie on the transverse axis, illustrating the expansive nature of the hyperbola as it approaches the foci.

Asymptotes of Hyperbola

Asymptotes are straight lines that a hyperbola approaches but never touches, providing a simplified guide to the hyperbola's behavior at infinity.
For any hyperbola, the equations of the asymptotes can be derived using the formula:\[ y - k = \pm \left(\frac{b}{a}\right)(x - h) \]Using this formula ensures a clear understanding of how the branches of a hyperbola align asymptotically.

Substituting our values \((h, k) = (1, 2)\), \(a = 1\), and \(b = 1\), we find:

• The asymptote equations are: \(y = x + 1\) and \(y = -x + 3\).

The equations indicate that our hyperbola's branches open towards these two non-intersecting diagonal lines. Observing these lines on a graph is essential, as they give a clear frame of reference for sketching the hyperbola accurately.