Problem 18 Sketch the graphs of the equatio... [FREE SOLUTION]

Chapter 10: Problem 18

Sketch the graphs of the equations given; indicate centers, foci. and asymptotes. $$x^{2}-y^{2}-2 x+6 y=9$$

Short Answer

Expert verified

The hyperbola is centered at (1, 3) with foci at (1卤3鈭�2, 3) and asymptotes y = x + 2 and y = -x + 4.

Step by step solution

Rewrite the Equation

The given equation is $x^2 - y^2 - 2x + 6y = 9$. We begin by rearranging terms to help complete the square for both $x$ and $y$. The equation can be rewritten as $(x^2 - 2x) - (y^2 - 6y) = 9$.

Complete the Square for x

Complete the square for the $x$ terms. Take $x^2 - 2x$, add and subtract $1$ (since $(2/2)^2 = 1$): $(x^2 - 2x + 1 - 1) = (x - 1)^2 - 1$.

Complete the Square for y

Complete the square for the $y$ terms. Take $y^2 - 6y$, add and subtract $9$ (since $(6/2)^2 = 9$): $(y^2 - 6y + 9 - 9) = (y - 3)^2 - 9$.

Simplify and Rearrange

Substitute the completed squares into the equation: $(x - 1)^2 - 1 - [(y - 3)^2 - 9] = 9$. Simplify to get: $(x - 1)^2 - (y - 3)^2 = 9$. This shows the graph of a hyperbola.

Find Center, Foci, and Asymptotes

The center of the hyperbola is $1, 3$. For a hyperbola $(x - h)^2/a^2 - (y - k)^2/b^2 = 1$, we have $a^2 = 9$ and $b^2 = 9$, so $a = 3$. The foci are $ (h \pm \sqrt{a^2 + b^2}, k) = (1 \pm \sqrt{18}, 3)$. The asymptotes are given by the equations $y - 3 = \pm \frac{3}{3}(x - 1)$, or $y = x + 2$ and $y = -x + 4$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square

The method of completing the square is a critical tool for transforming a quadratic expression into a form that is easier to analyze and interpret, especially when dealing with conic sections like hyperbolas.
The primary goal of completing the square is to express a quadratic polynomial in the form $(x - h)^2$ or $(y - k)^2$. This makes it easier to identify the center and other characteristics of conic sections.
Here鈥檚 a quick guide on how to complete the square:

Take the quadratic expression you want to transform, for example, $x^2 - 2x$.
Identify the coefficient of the linear term (the term that multiplies $x$); here, it is $-2$.
Find half of this coefficient and square it, i.e., $(2/2)^2 = 1$. This is the number you'll add and subtract within the expression.
Rewrite the expression involving this squared term, so it becomes $(x^2 - 2x + 1) - 1 = (x - 1)^2 - 1$.
Repeat this method for the $y$ terms as well.

By completing the square for both $x$ and $y$ in equations, you can rearrange them into standard forms that clearly show the structural properties of conics such as hyperbolas.

Foci of a Hyperbola

The foci of a hyperbola are crucial points that help define the shape and properties of the hyperbola. Unlike ellipses and circles, a hyperbola has two distinct foci.
These are located along the transverse axis at a distance described by the hyperbola's equation. In the mathematical form, the hyperbola鈥檚 equation is written as $(x - h)^2/a^2 - (y - k)^2/b^2 = 1$.
Here鈥檚 how you find the foci:

For the given hyperbola, calculate $a^2 + b^2$ to determine $c^2$, where $c$ is the distance from the center to each focus.
Solve for $c$ as $c = \sqrt{a^2 + b^2}$.
Plug the values into $(h \pm c, k)$ to find the coordinates of the foci.

In our equation, since $a^2 = 9$ and $b^2 = 9$, $c$ is found by calculating $\sqrt{18}$, and the foci become $(1 \pm \sqrt{18}, 3)$. The foci are two critical marks ensuring that any point on the hyperbola maintains a constant absolute distance difference to these points.

Asymptotes of a Hyperbola

Asymptotes give a hyperbola its distinctive open shape by describing lines that the hyperbola approaches but never actually touches or crosses. Understanding asymptotes allows you to sketch the graph of a hyperbola more accurately.
The equations of the asymptotes for a hyperbola centered at $(h, k)$ and given in standard form $(x - h)^2/a^2 - (y - k)^2/b^2 = 1$ are derived from the slopes $\pm b/a$.
To find the asymptotes:

Calculate the slope $b/a$; this is simply the ratio of $b$ over $a$.
Use the point-slope form of a line equation, starting from $y - k = \pm (b/a)(x - h)$.
Find the specific lines that serve as the asymptotes by substituting $a$ and $b$ values into the formula.

For the problem at hand, since both $a$ and $b$ equal 3, the slopes are $\pm 1$. Thus, the equations of the asymptotes are $y = x + 2$ and $y = -x + 4$. Asymptotes help define the direction in which each arm of the hyperbola extends indefinitely.

91影视

Sketch the graphs of the equations given; indicate centers, foci. and asymptotes. $$x^{2}-y^{2}-2 x+6 y=9$$

Short Answer

Step by step solution

Rewrite the Equation

Complete the Square for x

Complete the Square for y

Simplify and Rearrange

Find Center, Foci, and Asymptotes

Key Concepts

Completing the Square

Foci of a Hyperbola

Asymptotes of a Hyperbola

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