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Math 1060 ~ Trigonometry
Learning Objectives In this section you will: A (^) B b^ C a c Hypote nuse Opp osit e Adjacent θ 28 Conic Sections: Hyperbolas
- Define a hyperbola in a plane.
- Determine whether an equation represents a hyperbola or some other conic section.
- Graph a hyperbola from a given equation.
- Determine the center, vertices, foci and eccentricity of a hyperbola.
- Find the equation of a hyperbola from a graph or from stated properties. Ex 1 : Given the points F 1 (- 5 , 0 ) and F 2 ( 5 , 0 ), plot several points such that the difference of the distances from F 1 and F 2 to each point is 4. Draw the curve connecting the points. F 1 F 2
Hyperbolas General form: Ax^2 + By^2 + Cx + Dy + E = 0 , where A and B have opposite signs. Given: two points (foci) and a distance ( c ). Definition: A hyperbola is the set of all points in a plane such that for each point on the hyperbola, the difference of its distances from two fixed points is constant. Vocabulary Transverse axis Asymptotes Center Foci Standard Form of an Equation of a Hyperbola with Center at ( 0 , 0 ) The variables a, b and c have a special relationship.
Ex 5 : Write an equation and sketch each of these. a) The hyperbola such that the center is (- 2 , 3 ), one of the asymptotes passes through ( 1 , 4 ) and it is vertically oriented. b) A hyperbola with vertices at (- 4 , 3 ) and ( 2 , 3 ) and foci at (- 6 , 3 ) and ( 4 , 3 ) Ex 6 : Write this equation in standard form, sketch it, including the foci. x^2 - 9 y^2 - 4 x - 18 y - 14 = 0 Eccentricity of a Hyperbola e = c/a
Ex 7 : Identify each of these equations as one of these: C - Circle E - Ellipse that is not a circle (longer in which direction) H - Hyperbola (facing which way) P - Parabola (facing which way) i) 9 x^2 - 4 y^2 - 36 x + 8 y - 4 = 0 ii) y^2 + 4 x - 2 y - 11 = 0 iii) 16 x^2 + 16 y^2 + 64 x - 32 y - 176 = 0 iv) - 9 x^2 + 25 y^2 - 54 x - 50 y - 281 = 0 v) 9 x^2 + 4 y^2 - 18 x + 16 y - 11 = 0 vi) x^2 - 6 x + 8 y - 7 = 0 vii) 2 x^2 + 3 y^2 + 12 x + 24 y + 60 = 0
