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Chapter 7: Problem 60

How can you distinguish an ellipse from a hyperbola by looking at their equations?

Short Answer

Expert verified
The key difference between an ellipse and a hyperbola when looking at their equations is the sign between the two terms. An ellipse always has a positive (+) sign, while a hyperbola always has a negative (-) sign.

Step by step solution

01

Standard Equations of an Ellipse and Hyperbola

The general equation of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a > 0\) and \(b > 0\). The standard equation of a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).
02

Identify the Distinguishing Feature

Having looked at the equations, it can be observed that the distinguishing feature between an ellipse and a hyperbola is the sign between the two terms of the equation. In the ellipse, the sign is positive (+) whereas, in the hyperbola, it is negative (-). So, this is the key feature when distinguishing an ellipse from a hyperbola by using their equations.

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Most popular questions from this chapter

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

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$x^{2}+4 y^{2}+10 x-8 y+13=0$$

Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length \(12 ;\) length of minor axis \(=6 ;\) center: \((0,0)\)

In Exercises 43-50, 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. $$x^{2}-y^{2}-2 x-4 y-4=0$$

Find the standard form of the equation of each ellipse satisfying the given conditions. $$\text { Foci: }(-2,0),(2,0) ; \text { vertices: }(-6,0),(6,0)$$
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