This document defines and provides key details about the equation of a hyperbola. It states that a hyperbola is the set of all points where the absolute difference between the distance to two fixed points, called foci, is a constant. The document outlines that a hyperbola has: 1) A center point equidistant between the two foci 2) Vertices that are the points where the hyperbola intersects the line between the foci 3) Asymptotes that are lines extending from the center point and serve as a guide for graphing the hyperbola.

1. HYPERBOLAINTRODUCTION
PRECALCULUS
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2. Definition and Equation of a Hyperbola

3. Definition and Equation of a Hyperbola

4. Definition and Equation of a Hyperbola

5. Definition and Equation of a Hyperbola
 Let F1 and F2 be two distinct points. The set of all
points P, whose distances from F1 and from F2 differ
by a certain constant, is called a hyperbola. The
pointsF1 and F2 are called the foci of the hyperbola.

6. Definition and Equation of a Hyperbola
 In the Figure, given are two points on the x-axis,
F1(c,0) and F2(c,0), the foci, both c units away from
their midpoint (0,0). This midpoint is the center of the
hyperbola. Let P(x,y) be a point on the hyperbola, and
let the absolute value of the difference of the
distances of P from F1 and F2, be 2a. Thus, |PF1-
PF2|=2a,

7. Definition and Equation of a Hyperbola
 Thus, |PF1- PF2|=2a,
 Algebraic manipulations allow us to rewrite this into
the much simpler

8.  (1) center: origin (0 ,0)
 (2) foci: F1(-c,0) and F2(c,0)
• Each focus is c units away from the center.
• For any point on the hyperbola, the absolute value of the difference
of its distances from the foci is 2a.
 (3) vertices: V1(-a,0) and V2(a,0)
• The vertices are points on the hyperbola, collinear with the center
and foci.
• If y = 0, then x =±a. Each vertex is a units away from the center.
• The segment V1V2 is called the transverse axis. Its length is 2a.

9.  (4) asymptotes: y = bx/a and y =bx/a, the lines L1 and L2
 • The asymptotes of the hyperbola are two lines passing through the
center which serve as a guide in graphing the hyperbola:
 • An aid in determining the equations of the asymptotes: in the
standard equation, replace 1 by 0, and in the resulting equation x2/
a2-y2/b2 = 0, solve for y.
 • To help us sketch the asymptotes, we point out that the
asymptotes L1 and L2 are the extended diagonals of the auxiliary
rectangle. This rectangle has sides 2a and 2b with its diagonals
intersecting at the center C. Two sides are congruent and parallel to
the transverse axis V1V2. The other two sides are congruent and
parallel to the conjugate axis, the segment shown which is
perpendicular to the transverse axis at the center, and has length 2b.

10. TRY THIS!
Example 1: Determine the foci, vertices, and asymptotes of
the hyperbola with equation
𝑥2
9
−
𝑦2
7
= 1. Sketch the graph,
and include these points and lines, the transverse and
conjugate axes, and the auxiliary rectangle.
Example 2. Find the (standard) equation of the hyperbola
whose foci are F1(5,0) and F2(5,0), such that for any point on
it, the absolute value of the difference of its distances from
the foci is 6.

11. THANK YOU
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