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Chapter 8: Problem 7

For the following exercises, determine which conic section is represented based on the given equation. $$x^{2}-10 x+4 y-10=0$$

Short Answer

Expert verified
The conic section is a vertical parabola.

Step by step solution

01

Rewrite the Equation

The given equation is \(x^2 - 10x + 4y - 10 = 0\). We should start by rearranging the terms to make it easier to identify the conic section. Let's group \(x\)-terms and \(y\)-terms: \(x^2 - 10x = -4y + 10\).
02

Complete the Square for x

We need to complete the square for the \(x\) terms. Take the coefficient of \(x\), which is -10, divide by 2 to get -5, and then square it to get 25. Add and subtract 25 inside the equation: \((x^2 - 10x + 25) = -4y + 10 + 25\). Thus, \((x-5)^2 = -4y + 35\).
03

Rearrange to Standard Form

Rewrite the equation into the standard form. We have \((x-5)^2 = -4(y - \frac{35}{4})\). Observe this form closely; it resembles the standard form of a parabola \((x-h)^2 = 4p(y-k)\), where vertex is \((h, k)\).
04

Identify the Conic Section

Since the equation now fits the form \((x-h)^2 = 4p(y-k)\), it is a vertical parabola. This is because there is only one squared term \(x^2\), and it matches the pattern for 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 specific type of conic section that you can identify by its unique shape. It only has one squared variable in its equation.
A key feature of the parabola is its symmetry; if you were to fold it in half along its axis, both sides would match perfectly.
  • The basic shape of a parabola can be visualized like a U or an upside-down U.
  • The point where the parabola makes its sharpest turn is called the vertex.
  • It has an axis of symmetry, which is a vertical line that runs through the vertex.
Parabolas occur frequently in mathematics because they represent quadratic equations. Whenever you are trying to identify a conic section as a parabola, check if there is one term squared and the squared term is isolated on one side of the equation.
Completing the Square
Completing the square is a useful technique for rewriting quadratic expressions. It helps in transforming standard quadratic equations to reveal important features like vertex easily.
This method works by turning a trinomial into a perfect square trinomial.
Here's how you can complete the square step-by-step:
  • Start with an equation in the form of \(ax^2 + bx + c\).
  • Focus on the \(x\) terms: take the coefficient of \(x\), divide it by 2, and square the result.
  • Add and subtract this square to the equation.
  • Rewrite the trinomial as a binomial squared \((x+d)^2\).
This helps in easily converting the equation into a form where characteristics like the vertex of a parabola are more easily observable, while also aiding in simplifying and analyzing parabolic paths.
Standard Form of a Parabola
The standard form of a parabola helps in quickly identifying and studying its properties. In coordinates, the standard form for a vertical parabola is given by \((x-h)^2 = 4p(y-k)\).
For a horizontal one, it is \((y-k)^2 = 4p(x-h)\).
In these forms, here's what each symbol means:
  • \(h, k\): These are the coordinates of the vertex, which is the turning point of the parabola.
  • \(p\): This value determines how "steep" or "wide" the parabola is. It represents the distance from the vertex to the focus.
  • The value of \(k\) changes how the graph moves up and down (for vertical parabolas) or side to side (for horizontal parabolas).
The conversion to this form allows for easy reading of important properties like direction, width, and vertex location. Recognizing these features at a glance simplifies the process of graphing and analyzing parabolas.

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Most popular questions from this chapter

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{2}{5-3 \sin \theta} $$

Graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r(3-2 \sin \theta)=6 $$

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-4 ; e=5\)

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{2}{1-\cos \theta} $$

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=\frac{2}{5} ; e=\frac{7}{2}\)
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