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Chapter 10: Problem 7

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples 3-5). $$ y^{2}-5 x-4 y-6=0 $$

Short Answer

Expert verified
The conic section is a parabola.

Step by step solution

01

Rearrange the Equation

Start with the given equation: \( y^2 - 5x - 4y - 6 = 0 \). Reorganize it to make it easier to complete the square. This involves having all the terms related to \( y \) on one side. Rearrange the equation as \( y^2 - 4y = 5x + 6 \).
02

Complete the Square

Take the \( y \)-related terms from the rearranged equation, \( y^2 - 4y \). To complete the square, take half of the coefficient of \( y \) (which is -4), square it (\((-4/2)^2 = 4\)), and add that inside the equation. Thus, add and subtract 4 on the left side: \( y^2 - 4y + 4 - 4 = 5x + 6 \). This becomes \( (y - 2)^2 - 4 = 5x + 6 \).
03

Simplify the Equation

Move the -4 to the right-hand side to simplify: \( (y - 2)^2 = 5x + 10 \).
04

Identify the Conic Section

Recognize the standard form of the equation. The equation \( (y - 2)^2 = 5(x + 2) \) is in the form \( (y-k)^2 = 4p(x-h) \), which represents a parabola. Thus, the conic section represented by the equation is a parabola.

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Key Concepts

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

Parabola
A parabola is a type of conic section you'll often encounter in mathematics. It looks like a U-shape or an inverted U. The most important things to know about parabolas are:
  • They have a vertex, which represents the turning point or the "tip" of the U-shape.
  • Parabolas are symmetrical. This means one side is a mirror image of the other.
  • Their equation form is either \( y^2 = 4px \) or \( x^2 = 4py \), depending on their orientation 鈥� whether they open up, down, left, or right.
When solving problems, like finding out if a given equation represents a parabola, look for this standard form. You can recognize the parabola by matching your equation to its form. In the equation provided in the exercise, you rearranged terms to see it clearly as a parabola because it fit the pattern \( (y-k)^2 = 4p(x-h) \). Understanding this equation template helps you predict the shape and direction of the parabola.
Completing the Square
Completing the square is a useful mathematical technique used to simplify quadratic equations. It helps to transform quadratic equations into a form that makes them easy to solve or to identify conic sections.Here鈥檚 how to do it:
  • Start with a quadratic expression, like \( y^2 - 4y \).
  • Find the coefficient of the linear term, here it鈥檚 -4, take half of it, square it, and then add and subtract that square from the equation. For example, half of -4 is -2, and \( (-2)^2 = 4 \).
  • Re-write the quadratic part of the expression with your new term, so \( y^2 - 4y + 4 \) and subtract 4 to keep the equation balanced. Now it becomes \( (y - 2)^2 - 4 \).
This method helps us recognize and work with the standard equations of conics, especially parabolas. In the exercise, completing the square was crucial to identify and simplify the equation to reveal the parabola form.
Equations of Conics
Conic sections include shapes like circles, ellipses, parabolas, and hyperbolas. Each has its own special set of equations.Here's a quick guide:
  • Circles: Their equation is \( (x-h)^2 + (y-k)^2 = r^2 \).

  • Ellipses: Their form resembles circles but stretched, \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).

  • Parabolas: Discussed earlier, they follow \( (y-k)^2 = 4p(x-h) \) or \( (x-h)^2 = 4p(y-k) \).

  • Hyperbolas: Look like two mirrored curves, with an equation such as \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).
The arrangement of terms and the corresponding values in each equation help you determine which conic section you are dealing with. In your exercise, after completing the square, you could clearly see it matched the parabola equation form. Understanding these equations not only aids in identifying conics but also in graphing and solving related problems. It鈥檚 an essential skill for geometry and algebra.

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Most popular questions from this chapter

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). \(16 x^{2}+4 y^{2}=32\)

Find the equation of the parabola with vertex at the origin and axis along the \(x\) -axis if the parabola passes through the point \((3,-1) .\) Make a sketch.

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Find the equation of the tangent line to the given curve at the given point. \(\frac{x^{2}}{2}-\frac{y^{2}}{4}=1\) at \((\sqrt{3}, \sqrt{2})\)

Name the conic (horizontal ellipse, vertical hyperbola, and so on ) corresponding to the given equation. \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\)
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