11.3 The Hyperbola, Exercises of Astronomy

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11.3 The Hyperbola

OBJECTIVES:

• Recognize the equation of a hyperbola.
• Graph hyperbolas by using asymptotes.
• Identify conic sections by their

equations.

Written by: Cindy Alder

The Hyperbola

• A hyperbola is formed when a plane cuts the cone at an angle closer to the axis than the side of the cone.
• The definition of a hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points, called the foci , remains constant.

Hyperbolas in Physical Situations

• A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.

Equation of a Hyperbola Centered

at the Origin

Equation Graph Description Identification

2 2 2 2 1

x y a b

 

2 2 2 2 1

y x b a

 

The 𝑥-intercepts are (𝑎, 0) and (−𝑎. 0). The asymptotes are found from (𝑎, 𝑏), 𝑎, −𝑏 , (−𝑎, −𝑏), and (−𝑎, 𝑏).

𝑥^2 has a positive coefficient. 𝑦^2 has a negative coefficient.

The 𝑦-intercepts are 0, 𝑏 and (0, −𝑏). The asymptotes are found from 𝑎, 𝑏 , 𝑎, −𝑏 , (−𝑎, −𝑏), and (−𝑎, 𝑏).

𝑦^2 has a positive coefficient. 𝑥^2 has a negative coefficient.

Graphing Hyperbolas

• Graph:

2 2 1 81 64

x y  

Graphing Hyperbolas

• Graph: 35 y^2  4 x^2  140

Equation Graph Description Identification

y  a x   h ^2  k

x  a y   k ^2  h

Parabola opens up if a > 0, down if a < 0. The vertex is ( h, k ). The axis is x = h.

It has an x^2 term and y is not squared. Parabola opens to the right if a > 0, to the left if a < 0. The vertex is ( h, k ). The axis is y = k.

It has a y^2 term and x is not squared.

2 2 2 2 1

x y a b

 

^ x^ ^ h^  2 ^  y^ ^ k^ ^2  r^2

The center is ( h, k ) and the radius is r.

x^2 and y^2 terms have the same positive coefficients. The x -intercepts are ( a , 0 ) and ( - a, 0 ). The y -intercepts are ( 0, b ) and ( 0 , -b ).

x^2 and y^2 terms have different positive coefficients. 2 2 2 2 1

x y a b

 

2 2 2 2 1

y x b a

 

The x -intercepts are ( a , 0 ) and ( - a, 0 ). The asymptotes are found from ( a , b ), ( a, -b ), ( - a, -b ), and ( - a, b ).

x^2 has a positive coefficient. y^2 has a negative coefficient. The y -intercepts are ( 0, b ) and ( 0, - b ). The asymptotes are found from ( a , b ), ( a, -b ), ( - a, -b ), and ( - a, b ).

y^2 has a positive coefficient. x^2 has a negative coefficient.

Identifying Conic Sections from

Their Equations

• Identify the graph of each equation.
  • 9𝑥^2 = 108 + 12𝑦^2

• 𝑥^2 = 𝑦 − 3

• 𝑥^2 = 9 − 𝑦^2

• 3𝑥^2 = 27 − 4𝑦^2

• 6𝑥^2 = 100 + 2𝑦^2

• 3𝑥^2 = 27 − 4𝑦