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Chapter 10: Problem 67

What is a hyperbola?

Short Answer

Expert verified
A hyperbola is a type of conic section defined by a specific equation and is the set of all points in a plane such that the difference of the distances from two fixed points is constant. It has two disconnected curves that open either right and left or up and down.

Step by step solution

01

Definition

A hyperbola is a type of conic section, defined by its specific equation. It is the set of all points in a plane such that the difference of the distances from two fixed points (the foci) is constant for all points.
02

Description of Shape

A hyperbola is a mirror image of itself across a central line, called the axis of the hyperbola. It is composed of two disconnected curves that open either up and down or right and left.
03

Equation

The standard form equation of a hyperbola is given by \( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \) if the hyperbola is oriented horizontally, or \( \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \) if it is oriented vertically, where (h, k) is the center of the hyperbola, and a and b are constants that determine the shape of the hyperbola.

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Most popular questions from this chapter

In Exercises \(51-60,\) 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. $$ 4 x^{2}+25 y^{2}-24 x+100 y+36=0 $$

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