conics written by gaurav rao last edited: 10/3/15
TRANSCRIPT
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ConicsWritten by Gaurav Rao
Last edited: 10/3/15
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What Are Conics?• Conics are cross sections of a cone
• Locus of points that distance from a point (focus) And a line (directrix) are at a fixed ratio(eccentricity)
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Parts of conic
• All conics have a Focus, and all conics but circles have diretricies• The eccentricity, defined as the ratio of the distances between the
focus and the directrix, is different for each type of conic
e Conic
e=0 Circle
0<e<1 Ellipse
e=1 Parabola
e>1 Hyperbola
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Identifying a conic
• All conics can be written in the form • The discriminant of the conic is (looks familiar?)• If the discriminant is positive, then the conic is a hyperbola• If the discriminant is zero, then it is a parabola• If the discriminant is negative, then it is an ellipse• If then it is a circle
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Ellipses
• Defined as the locus of points on a plane whose sum of the distances to two points is constant• Focal Length (c) • Eccentricity • Directrix • Latus rectum • Area
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Hyperbolas
• Locus of points on a plane whose differences of the distances between two points is constant
• Focal Length (c) • Eccentricity • Directrix • Latus rectum
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Parabolas
• Locus of points at an equal distance from a point an a line
• Focal length = p• Latus rectum 4p• All parabolas have a eccentricity of 1
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Rotation of axis
• If there is a B term in the standard form of a conic, then there is a rotation. • To find the angle, find θ in this equation:
• Then Substitute in and
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Degenerate Conics
• Hyperbolae can degenerate into two intersecting lines, two parallel lines• Parabolae can degenerate to two parallel lines or a double line• Ellipses can degenerate to two parallel lines or the double line• If the determinant of the following matrix is zero, then the conic is
degenerate
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Practice problems
• Identify the following non-degenerate conic by using the discriminant:
• A) Hyperbola B) Circle C) Non-circular Ellipse D) Cycloid E) NOTA
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Practice Problem 2
• James graphed the following equation on the Argand diagram: . What conic shape did James just graph?
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Practice Problem 3
• Find the area and eccentricity of the following conic