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Chapter 7: Problem 55

What is a hyperbola?

Short Answer

Expert verified
A hyperbola is a type of conic section defined as the set of all points in a plane whose distance from two distinct fixed points (the foci) is a positive constant. It consists of two mirrored branches with unique features such as asymptotes and eccentricity.

Step by step solution

01

Understanding Hyperbola

A hyperbola is one of the four types of conic sections, the others being circle, ellipse, and parabola. It is defined as the set of all points (x,y) in a plane, the difference of whose distances from two distinct fixed points (the foci) is a positive constant.
02

Characteristics of Hyperbola

It consists of two identical halves, called 'branches', which mirror each other across the center. At the center, the curve bends the sharpest, and this bend corresponds to the vertices of the hyperbola.
03

Noteworthy Features of Hyperbola

Two unique features of a hyperbola are its asymptotes and the eccentricity. Asymptotes are diagonal lines that the branches of hyperbola approach but never reach. The eccentricity measures how 'un-circular' the hyperbola is, it is always greater than 1.

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Most popular questions from this chapter

Find the standard form of the equation of each ellipse satisfying the given conditions. $$\text { Foci: }(0,-3),(0,3) ; \text { vertices: }(0,-4),(0,4)$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$4 y^{2}=1-4 x^{2}$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$x^{2}=1-4 y^{2}$$

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$\frac{(y-2)^{2}}{36}-\frac{(x+1)^{2}}{49}=1$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$6 x^{2}=30-5 y^{2}$$
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