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Chapter 13: Problem 38

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

Short Answer

Expert verified
An ellipse's equation has the x and y squared terms summed together, whereas a hyperbola's equation has them separated by a negative sign. Moreover, both terms in an ellipse are either under the same fraction or have the same associated number, while in a hyperbola they each have their own fraction.

Step by step solution

01

Identify term signs

Find the general equation of the given conic section. If all the terms are added together, then it's likely an equation for an ellipse. If there is subtraction involved, then we are likely dealing with a hyperbola.
02

Confirm the distinction between ellipse and hyperbola

Even when subtraction is involved, it might be a circle (if all the exponents are 2) or a parabola (if one term is missing). The presence of x squared and y squared terms (with different coefficients), separated by a negative sign, typically represents a hyperbola. In contrast, if these two terms are separated by a positive sign, that typically represents an ellipse.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I noticed that depending on the values for \(A\) and \(B\), assuming that they are not both zero, the graph of \(A x^{2}+B y^{2}=C\) can represent any of the conic sections other than a parabola.

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. Then graph the conic section. $$4 x^{2}+4 y^{2}=16$$

Explain how to use \(x=y^{2}+8 y+9\) to find the parabola's vertex.

The equation of a parabola is given. Determine: a. if the parabola is horizontal or vertical. b. the way the parabola opens. c. the vertex. $$x=-(y+1)^{2}+4$$

Find the slope of the line passing through \((-2,-3)\) and \((1,5)\).
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