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## Topics in Analytic Geometry

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**Topics in Analytic Geometry**Pre Calc Chapter 9**Parabolas**• Vertex • The lowest or highest point of the graph (based on which way it opens) • Axis of symmetry**Geometric Definition**• A parabola is the set of points in the plane equidistant from a fixed point F (the focus) and a fixed line l called the directrix**Analytic Geometry**• Concerned with shapes, not necessarily functionality • Parabolas can open up • Parabolas can open down • Inverse**Parabola with Vertical Axis**The Graph of the equation:is a parabola with the following: Vertex Focus Directrix Parabola opens up if or down if**Parabola with Horizontal Axis**The Graph of the equation:is a parabola with the following: Vertex Focus Directrix Parabola opens right if or left if**Focal Diameter**• Distance across the parabola along the line parallel to the directrix**Geometric Definition**• An ellipse is the set of all points in the plane the sum of whose distances from two fixed points and is a constant. • These points are the foci**Ellipse**Equation Vertices Major Axis Horizontal, length 2a Vertical, 2a Minor Axis Vertical, length 2b Horizontal, 2b Foci**Eccentricity**• For the ellipse or the eccentricity, e, is the numberwhere and the eccentricity of every ellipse satisfies**Ellipses**• Find the equation of the ellipse with foci and eccentricity**Geometric Definition**• A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points and is a constant. • These are the foci of the hyperbola**Hyperbolas**Equations Vertices Transverse Axis Horizontal, length 2a Asymptotes Foci**Hyperbolas**Equations Vertices Transverse Axis Vertical, length 2b Asymptotes Foci**Sketching Hyperbolas**• Sketch the Central Box • Sketch the asymptotes • Plot the vertices • Sketch the hyperbola • Smile **Hyperbolas**• Find the equation of the hyperbola with vertices and foci**General Equation of a Conic**• The graph of the equationWhere A and C are not both 0, is a conic or degenerate conic where the graph is: • A parabola if A or C is 0 • An ellipse if A and C have the same sign • Circle if A = C • A hyperbola is A and C have opposite signs**Degenerate Conic**• Conic which simplifies to only 2 lines**Rotation of Axes**• Recall… • Now…**Rotation of Axes Formula**• Suppose the x- and y-axes in a coordinate plane are rotated through the acute angle to produce the X- and Y-axes. Then the coordinates (x,y) and (X,Y) of a point in the xy- and XY-panes are related as follows: x=Xcosφ-Ysinφ X=xcosφ+ysinφ y=Xsinφ+Ycosφ Y=-xsinφ+ycosφ**Rotation of Axes**• If the coordinates are rotated 30 degrees, find the XY-coordinates of the point with xy-coordinates (2, -4)**Rotation of Axes**• Rotate the coordinate axes through 45 degrees to show that the graph of the equation xy = 2 is a hyperbola**Polar Coordinates**• Uses distances and directions to specify locations on the plane • Origin (Pole) • Polar Axis • Polar Coordinates