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Chapter 7: Problem 73

How can you distinguish parabolas from other conic sections by looking at their equations?

Short Answer

Expert verified
A parabola can be distinguished from other conic sections by looking at their equations. A parabola's equation only includes one square term ('x^2' or 'y^2'), but never both. It also never contains an 'xy' term, unlike other conic sections. It's a second degree equation in one variable only, unlike ellipses and hyperbolas that are second degree equations in both variables.

Step by step solution

01

Understanding the Properties of a Parabola

A parabola has an equation of the form \(y=ax^2+bx+c\) or \(x=ay^2+by+c\) where \(a\), \(b\), and \(c\) are constants. The graph of a parabola always has a vertex and an axis of symmetry.
02

Distinguishing Parabolas from Other Conic Sections

a) An ellipse (including circles) has an equation of the form \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) or \(\frac{y^2}{a^2}+\frac{x^2}{b^2}=1\). \ b) A hyperbola has an equation of the form \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) or \(\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\). If an equation matches to either of these forms, it's not a parabola. A crucial thing to note is that a parabolic equation has only one square power term, either \(x^2\) or \(y^2\), but never both as opposed to other conic sections.
03

Practice on Examples

Applying the rules to identify conic sections to various equations and determining their type would help cement the understanding. We should remember that parabolas do not have an 'xy' term while hyperbolas and ellipses do.

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Most popular questions from this chapter

Graph each relation. Use the relation's graph to determine its domain and range. \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\)

Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: \((2,-3) ;\) Focus: \((2,-5)\)

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(y+3)^{2}=12(x+1)$$

Will help you prepare for the material covered in the first section of the next chapter. Evaluate \(i^{2}+1\) for all consecutive integers from 1 to 6 inclusive. Then find the sum of the six evaluations.

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{c}4 x^{2}+y^{2}=4 \\\2 x-y=2\end{array}\right.$$
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