basicconicsectionsbasic conic sections...conicshistoryconics history • discosco e edvered dudu...
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Basic Conic SectionsBasic Conic SectionsJustin FilbertBerwick High School 9th Grade
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Conics HistoryConics History• Discovered during the classical Greek period (600‐sco e ed du g e c ass ca ee pe od (600300 B.C.)
• The Greeks focused mainly on the geometric properties
• In the early 17th century, the applicability was ti d d l d t l i th d l tnoticed and played a strong role in the development
of Calculus
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Basic ConicsBasic Conics• A conic section is the co c sect o s eintersection of a double‐napped cone d land a plane
• In forming the fourbasic conics the interbasic conics, the inter‐section of the plane and the cone does not pass through the vertex
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Basic Conics (cont )Basic Conics (cont.)•
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CirclesCircles•
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The Standard Equation of a Circle•
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Finding the Standard Equation of a Circle
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ParabolasParabolas•
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The Standard Equation of a Parabola•
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Finding the Standard Equation of a Parabola•
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Reflective Property of Parabolas• A line segment that passes through the focus of a g p gparabola and has endpoints on the parabola is called a focal chord
Th l t t i th ifi f l h d di l• The latus rectum is the specific focal chord perpendicular to the axis of the parabola
• A line is tangent is to a parabola at a point on the g p pparabola if the line intersects, but does not cross, the parabola at the point
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The EllipseThe Ellipse• An ellipse is defined as the set of all points (x,y) in a p p ( ,y)plane, the sum of whose distances from two distinct fixed points [foci (singular‐focus)] is constant
Th li th h th f i i t t th lli t t• The line through the foci intersect the ellipse at two points called vertices
• The major axis is the chord the joins the verticesj j
• The midpoint of the ellipse is called the center of the ellipse
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The Standard Equation of an Ellipse•
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Finding the Standard Equation of an Ellipse•
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Application of an EllipseApplication of an Ellipse• Ellipses have many practical and aesthetic uses, such as:
• Machine gears
• Supporting arches
• Acoustic designs• Acoustic designs
• The orbits of satellites and planets
• Etc.
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EccentricityEccentricity•
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Application of EccentricityApplication of Eccentricity•
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TheHyperbolaThe Hyperbola• A hyperbola can be defined as the set of all points (x y) in a place theof all points (x,y) in a place, thedifference of whose distance from two distinct fixed points, the foci, is a positive constant
• A hyperbola is similar to a ellipse:• A hyperbola is similar to a ellipse:• An ellipse is the sum of the distances
between the foci and a point on the ellipse is constant
• Whereas a hyperbola is the difference• Whereas, a hyperbola is the differenceof the distances between the foci and two distinct fixed points is constant
• The two disconnected parts of the graph of ahyperbola are called the brancheshyperbola are called the branches
• The line through the two foci intersect the hyperbola at two points called the vertices
• The line connecting the vertices is the transverse axisTh id i t f th t i th t f th h b l• The midpoint of the transverse is the center of the hyperbola
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The Standard Equation of a Hyperbola•
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The Standard Equation of a Hyperbola•
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Asymptotes of a HyperbolaAsymptotes of a Hyperbola•
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Finding the Asymptotes and Standard Form of a Hyperbola
•
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Finding the Asymptotes and Standard Form of a Hyperbola•
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General Equations of ConicsGeneral Equations of Conics•
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Further ResearchFurther Research•
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Thank You!Thank You!• I’d like to thank the judges and students for taking time to be here today
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ReferencesReferences