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Chapter 7: Problem 61

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

Short Answer

Expert verified
Parabolas can be distinguished from other conic sections by examining their equations; an equation that includes a single squared variable (either 'x' or 'y') typically represents a parabola.

Step by step solution

01

Familiarize with Conic Sections

Conic sections are curves obtained as the intersection of the surface of a cone with a plane. There are four types of conic sections: these include circle, ellipse, parabola, and hyperbola.
02

Understand the Standard Forms

Each conic section has a standard form. The general standard forms are: \n- Circle: \(x^2 + y^2 = r^2\) \n- Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) (for horizontal), or \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) (for vertical) \n- Parabola: \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\)\n- Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
03

Identify a Parabola

A parabola is the set of points that are an equal distance from a point (the focus) and a line (the directrix). In the equation of a parabola, only one variable is squared. This is the distinguishing feature of a parabola. So if you see an equation where only the 'x' or only the 'y' is squared, you are looking at a parabola. Identifying this key feature in the equation will help distinguish a parabola from other conic sections.

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Most popular questions from this chapter

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$4 x^{2}-9 y^{2}+8 x-18 y-6=0$$

Find the standard form of the equation of each ellipse satisfying the given conditions. $$\text { Foci: }(-5,0),(5,0) ; \text { vertices: }(-8,0),(8,0)$$

Use a graphing utility to graph the parabolas.Write the given equation as a quadratic equation in \(y\) and use the quadratic formula to solve for \(y .\) Enter each of the equations to produce the complete graph. $$y^{2}+2 y-6 x+13=0$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$16 x^{2}-y^{2}+64 x-2 y+67=0$$

Graph each ellipse and give the location of its foci. $$\frac{(x+2)^{2}}{16}+(y-3)^{2}=1$$
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