Problem 55
Question
What is a hyperbola?
Step-by-Step Solution
Verified Answer
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.
1Step 1: 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.
2Step 2: 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.
3Step 3: 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.
Other exercises in this chapter
Problem 54
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}+1
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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. $$4 x^{2}+y^{2}+1
View solution Problem 56
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. $$4 x^{2}+25 y^{2
View solution Problem 56
Describe how to graph \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\)
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