PRE-CAL | Hyperbolas

Definition

• set of all points, the difference of whose distances from two fixed points called foci is constant.

Properties

• Foci: the fixed points described in the definition above.

• Vertices: the points where the direction of the curves of the hyperbola changes.

• Transverse axis: the chord that passes through the vertices.

• Center: the midpoint of the transverse axis.

• Conjugate Axis: the chord that connects the co-vertices, is perpendicular to the transverse axis.

• Co-vertices: points that are b units away from the center.

• Asymptotes: the diagonal lines that the curves of hyperbola approach but never intersect.

Formula

Variables

• a is the distance from the center to a vertex

• 2a is the length of the transverse axis.

• b is the distance form the center to a co-vertex

• 2b is the length of the conjugate axis

• c is the distance from the center to the focus.

Pythagorean Relationship

• Create a rectangle such that the sides pass through the vertices and co-vertices.

• c is equal to the line segment connecting the center and a corner, creating part of the asymptotes. It forms the hypotenuse of a right triangle, where a and b are the sides.

• c² = a² + b²

Equations

General Equation

• |PF₁| - |PF₂| = 2a

  • PF is the distance between a point on the ellipse and the focus.

  • a is the distance from the center to a vertex.

Standard Equation for HORIZONTAL HYPERBOLAS

• x²/a² - y²/b² = 1

Standard Equation for VERTICAL HYPERBOLAS

• y²/a² - x²/b² = 1

Graphing a Hyperbola

1. Identify if the hyperbola is a horizontal or vertical hyperbola based on which variable is in the positive term.

2. Identify the center of the hyperbola, in this case is the origin.

3. Identify and graph the vertices by getting the square root of the denominator of the first term.

   • In a horizontal hyperbola, identify the coordinates of the endpoints by adding and subtracting this value from the x-coordinate of the center. The vertices should be at (a,0) and (-a,0).

   • In a vertical hyperbola, identify the coordinates of the endpoints by adding and subtracting this value from the y-coordinate of the center. The vertices should be at (0,a) and (0,-a).

4. Identify and graph the co-vertices by getting the square root of the denominator of the second term.

   • In a horizontal hyperbola, identify the coordinates of the endpoints by adding and subtracting this value from the y-coordinate of the center. The co-vertices should be at (0,b) and (0,-b).

   • In a vertical hyperbola, identify the coordinates of the endpoints by adding and subtracting this value from the x-coordinate of the center. The co-vertices should be at (b,0) and (-b,0).

5. Identify the value of c by getting the square root of a² + b² to plot the foci.

   • In a horizontal hyperbola, add and subtract the solved value from the x-coordinate of the center. The foci should be at (c, 0) and (-c, 0).

   • In a vertical hyperbola, add and subtract the solved value from the y-coordinate of the center. The foci should be at (0, c) and (0, -c).

6. Draw a rectangle such that the vertices and co-vertices are a midpoint of a side. Make the asymptotes by drawing two diagonal lines connecting the opposite corners of the rectangle

   • In a horizontal hyperbola, the equations of the asymptotes are y = ±b/a.

   • In a vertical hyperbola, the equations of the asymptotes are y = ±a/b.

7. (OPTIONAL) To make graphing easier, plot four more points by solving for the latus rectum.

   • Solve for the distance of the latus rectum with b²/a.

   • In a horizontal hyperbola, add and subtract the solved value from the y-coordinate of the foci. The points should be at (c, b²/a), (c, -b²/a), (-c, b²/a), and (-c, -b²/a).

   • In a vertical hyperbola, add and subtract the solved value from the x-coordinate of the foci. The points should be at (b²/a, c), (-b²/a, c), (b²/a, -c), and (-b²/a, -c).

8. Draw the curves of the hyperbola using the asymptotes as guides. Ensure that the curves approach the asymptotes but never meet.