Hyperbolas Sec 8 3 a Definition Hyperbola A

Hyperbolas Sec. 8. 3 a

Definition: Hyperbola A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola. How is this different from an ellipse? ? ?

Definition: Hyperbola Focus Vertex Center Focal Axis Focus

Deriving the Equation of a Hyperbola Notice: 0 Combining: Distance Formula:

Deriving the Equation of a Hyperbola

Deriving the Equation of a Hyperbola Let Divide both sides by

Deriving the Equation of a Hyperbola This equation is the standard form of the equation of a hyperbola centered at the origin with the x-axis as its focal axis. When the y-axis is the focal axis? Chord – segment with endpoints on the hyperbola Transverse Axis – chord lying on the focal axis, connecting the vertices (length = 2 a) Conjugate Axis – segment (length = 2 b) that is perp. to the focal axis and has the center of the hyperbola as its midpoint

Deriving the Equation of a Hyperbola This equation is the standard form of the equation of a hyperbola centered at the origin with the x-axis as its focal axis. When the y-axis is the focal axis? Semitransverse Axis – the number “a” Semiconjugate Axis – the number “b”

Deriving the Equation of a Hyperbola The hyperbola has two asymptotes, which can be found by replacing the “ 1” in the equation with a “ 0”: Solve for y Drawing Practice: Steps to sketching the hyperbola

Hyperbolas with Center (0, 0) • Standard Equation • Focal Axis • Foci • Vertices • Semitrans. Axis • Semiconj. Axis • Pythagorean Relation • Asymptotes x-axis y-axis

Hyperbolas with Center (0, 0)

Guided Practice Find the vertices and the foci of the hyperbola Sketch the hyperbola? Standard Equation: Vertices: Foci:

Guided Practice Find an equation of the hyperbola with foci (0, – 3) and (0, 3) whose conjugate axis has length 4. Sketch the hyperbola and its asymptotes, and support your sketch with a grapher. General Equation: c=3 b=2 a= 5 The Sketch? ? ? Standard Equation:

Let’s see some hyperbolas whose centers are not on the origin…

Hyperbolas with Center (h, k) • Standard Equation • Focal Axis • Foci • Vertices • Semitransverse Axis • Semiconjugate Axis • Pythagorean Relation • Asymptotes

Guided Practice Find the standard form of the equation for the hyperbola whose transverse axis has endpoints (– 2, – 1) and (8, – 1), and whose conjugate axis has length 8. Start with a diagram? General Equation: The center is the midpoint of the transverse axis:

Guided Practice Find the standard form of the equation for the hyperbola whose transverse axis has endpoints (– 2, – 1) and (8, – 1), and whose conjugate axis has length 8. Semitransverse Axis: Semiconjugate Axis: Specific Equation:

Guided Practice Find the center, vertices, and foci of the given hyperbola. The graph? Center: Vertices: Foci:

Guided Practice Find an equation in standard form for the hyperbola with transverse axis endpoints (– 2, – 2) and (– 2, 7), slope of one asymptote 4/3. Start with a graph? Center is the midpoint of the transverse axis: Find a, the semi-transverse axis: Asymptote slope is a/b:

Guided Practice Find an equation in standard form for the hyperbola with transverse axis endpoints (– 2, – 2) and (– 2, 7), slope of one asymptote 4/3. General equation: Plug in data: