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Chapter 11: Problem 25

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$x^{2}-4 y^{2}-2 x+16 y=20$$

Short Answer

Expert verified
The equation represents a hyperbola with center (1, 2).

Step by step solution

01

Rearrange Terms

Start by rearranging the equation to separate the terms involving \(x\) and \(y\):\[x^2 - 2x - 4y^2 + 16y = 20\]
02

Complete the Square for x

Focus on the \(x\) terms first: \(x^2 - 2x\). To complete the square, take half of the coefficient of \(x\), square it, and add and subtract it inside the equation:\[x^2 - 2x = (x-1)^2 - 1\]
03

Complete the Square for y

Now focus on the \(y\) terms: \(-4y^2 + 16y\). Factor out the \(-4\), then complete the square:\[-4(y^2 - 4y) = -4((y-2)^2 - 4) = -4(y-2)^2 + 16\]
04

Substitute Completed Squares Into Equation

Replace the expressions into the original equation and simplify:\[(x-1)^2 - 1 - 4(y-2)^2 + 16 = 20\]Simplify the equation:\[(x-1)^2 - 4(y-2)^2 = 5\]
05

Identify the Conic Section

The equation is now in the form \[(x-1)^2 - 4(y-2)^2 = 5\], which matches the standard form of a hyperbola \[(x-h)^2/a^2 - (y-k)^2/b^2 = 1\]. Thus, it represents a hyperbola.
06

Convert to Standard Form of a Hyperbola

Divide the entire equation by 5 to express the standard form:\[\frac{(x-1)^2}{5} - \frac{(y-2)^2}{5/4} = 1\]
07

Identify Hyperbola Features

From the equation \[\frac{(x-1)^2}{5} - \frac{(y-2)^2}{5/4} = 1\], identify:- Center: \((1, 2)\)- Vertices: Horizontally away from center \((1\pm\sqrt{5}, 2)\)- Foci calculation: \(c = \sqrt{a^2 + b^2} = \sqrt{5 + \frac{5}{4}}\)- Asymptotes: Lines passing through \((1, 2)\) with slopes \(\pm\frac{b}{a}\).
08

Calculate Foci and Asymptotes

The slopes of the asymptotes are determined by \(b/a = \sqrt{\frac{5}{4}}/\sqrt{5}\), and the foci distance \(c = \sqrt{\frac{25}{4}} = \frac{5}{2}\). Hence, foci are located at \((1\pm\frac{5}{2}, 2)\), and the asymptote equations are \(y = 2 \pm \frac{x-1}{2}\).

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Key Concepts

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

Completing the Square
Completing the square is a key algebraic technique used to simplify a quadratic expression. It is particularly useful when dealing with conic sections because it helps to rewrite quadratic equations into standard forms. This makes it easier to identify and graph these curves. When completing the square, follow these steps:
  • Identify the quadratic term (e.g., the coefficient of x or y squared).
  • Take half of the linear coefficient, square it, and add it within the expression.
  • If the equation requires, remember to balance both sides by adding or subtracting the same value.
For example, given a term like \(x^2 - 2x\), take half of \(-2\), which is \(-1\), square it to get \(1\), and adjust: \((x-1)^2 - 1\). This transformation allows us to achieve a perfect square trinomial, facilitating the analysis of the conic section's properties.
Hyperbola
A hyperbola is one of the intriguing shapes in the family of conic sections, characterized by its two separate, mirror-image curves. In a hyperbola, you will notice the distinguishing presence of two squared terms subtracted from each other. The standard equation of a hyperbola is given by:\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]Here,
  • \((h, k)\) represents the center of the hyperbola.
  • \(a\) defines the distance from the center to the vertices.
  • \(b\) relates to the distance determining the slop of asymptotes in a hyperbola鈥檚 graph.
These distinct features make it crucial to differentiate a hyperbola from other conic sections, especially when solving equations. Notice how the subtraction between squared terms highlights this particular form.
Conic Section Identification
Conic sections are curves obtained by slicing a double-napped cone with a plane in various ways. They include ellipses, parabolas, hyperbolas, and circles. Identifying conic sections involves observing whether each squared term is added or subtracted in the equation:
  • If both are added, we possibly have an ellipse or a circle.
  • If they are subtracted, as in \((x-h)^2 - (y-k)^2 = 1\), you are most likely dealing with a hyperbola.
Recognition of these patterns is fundamental to deciding which conic section the equation portrays and further analysis for determining specific details such as vertices, foci, and axes lengths follows logically from this identification.
Graphing Conic Sections
Graphing conic sections involves piecing together the various insights gained from equation manipulation (like completing the square) and conic characteristic identification. Steps to sketching a hyperbola, for instance include:
  • Placing the center point from coordinates \((h, k)\).
  • Plotting vertices by moving distance \(a\) from the center.
  • Drawing asymptotes, which cross the center, guiding the curve's direction.
  • Calculating the foci, extending further from center at distance \(c\), where \(c^2 = a^2 + b^2\).
The hyperbola鈥檚 curves approach but never touch the asymptotes, sketching these guidelines in the graph highlights the structure and opens a window into conic sections鈥� beauty. Always align your plots with your calculated and identified features from the equation.

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Most popular questions from this chapter

A polar equation of a conic is given. (a) Show that the conic is a hyperbola, and sketch its graph. (b) Find the vertices and directrix, and indicate them on the graph. (c) Find the center of the hyperbola, and sketch the asymptotes. $$r=\frac{8}{1+2 \cos \theta}$$

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((0, \pm 1),\) length of transverse axis: 1

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{2}{1-\cos \theta}$$

Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. $$\left\\{\begin{aligned} 100 x^{2}+25 y^{2} &=100 \\ x^{2}+\frac{y^{2}}{9} &=1 \end{aligned}\right.$$

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{8}{3+\cos \theta}$$
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