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Chapter 10 Conics and Polar Coordinates

• 10.1 The Parabola
  • Directrix (l)
  • Focus (F)
  • Eccentricity (e)
    • 0<e<1 Ellipse
    • e=1 Parabola
      • Standard equation
    • e>1 Hyperbola
  • Exercises
    • Find the focus and directrix of parabola
      • 1 Determine the standard type of parabola
      • 2 Find p
      • 3 Write down the focus and directrix based on p
    • Find the equation of parabola that go through a point and opens direction
      • 1 Determine the standard type of parabola
      • 2 Find p by using point
      • 3 Write down the equation based on p
• 10.2 Ellipses and Hyperbolas
  • Ellipse
    • Standard equation
      • Focus on x-axis
        • Focus (c,0) and (-c,0)
        • Vertex (a,0) , (-a,0) and (0,b) , (0,-b)
      • Focus on y-axis
        • Focus (0,c) and (0,-c)
        • Vertex (0,a) , (0,-a) and (b,0) , (-b,0)
  • Hypobola
    • Standard equation
      • Focus on x-axis
        • Asymptotes
        • Vertex (a,0) and (-a,0)
      • Focus on y-axis
        • Asymptotes
        • Vertex (0,a) and (0,-a)
  • Exercises
    • Find the focus and eccentricity and asyptotes
      • 1 Determine the major axis
      • 2 Find a and b
      • 3 Calculate c
      • 4 Write down the focus and eccentricity based on a,b,c
    • Find the equation
    • Find the area of ellipse
      • S=2Supper
• 10.3 Translation and Rotation of Axes
  • Quadriatic equation
  • Translations
    • If the new axes are parallel, respectively, to the original axes and have the same directions and scales
    • (x,y) old xy-coordinate system (u,v) new uv-coordinate system Let (h,k) be the old coordinates of the new origin
    • u=x-h v=y-k
  • Rotations
    • If the equation includes the cross product term Bxy
    • (x,y) old xy-coordinate system (u,v) new uv-coordinate system
    • x=ucosα-vsinα y=vcosα+usinα
  • Exercises
    • Find the new euqation after coordinates translations
      • 1 Find h and k
      • 2 Calculate u and v
      • 3 Replace x and y by u and v, respectively
    • Find the new coordinates of a point after a translation of axes to a new oringin
      • 1 Find h and k which is the coordinates of new origin
      • 2 Calculate u and v
      • 3 (u,v) is the new coordinates of the point
    • Find new equation after rotation of axes through α
      • 1 Substituting x=ucosα-vsinα y=vcosα+usinα
      • 2 Replace x and y by the equation in step 1
    • Make a rotation of axes to eliminate the cross-product term in the equation
      • 1 Determine the angle α by using the formula Bcos2α=（A-C）sin2α
      • 2 Substituting x=ucosα-vsinα y=vcosα+usinα
      • 3 Replace x and y by the equation in step 2
• 10.7 Calculus in Polar Coordinates
  • Area in Polar coordinates
  • Tangent line for polar curves
    • • • the tangent at the pole
      • • the slope of the tangent line
  • Arc length for polar curves
  • Exercises
    • Find the area of the region
      • 1 Find dA
      • 2 Find A
    • Find the area of region outside the curve f(α) and inside the curve g(α)
      • 1 Solving the two equations simultaneously
      • 2 Calculate
    • Find the slope of tangent line at α
      • 1 Find
      • 2 Substituting the value of α
    • Find the tangent lines at the pole
      • 1 Set the polar equation f(α)=0 and solve
      • 2 Find tanα
      • 3 The tangent lines at the pole in rectangular coordinate system are y=tanαx
    • Find the perimeter of the curve f(α)
      • 1 Transfrom polar equation to rectangular equation
      • 2 Calculate
• 10.6 Graphs of Polar Equations
  • Type 1
    • r=a Circle
  • Type 2
    • α=α0 Line
  • Type 3
    • r=2acosα Circle with a diameter lies in the x-axis
  • Type 4
    • r=2asinα Circle with a diameter lies in the y-axis
  • Type 5
    • r=aα Archimedean Spiral
  • Type 6
    • Cardioid Limacon
  • Type 7
    • Lemniscate
  • Type 8
    • r=asinnα or r=acosnα Rose Curves
  • Exercises
    • Identity the graph of polar equation
• 10.5 The Polar Coordinate System
  • Polar coordinate system
    • A point O called Pole (or origin)
    • A ray Ox called Polar axis
    • P is any point in the plane, r is the distance from O to P and let α be the angle from polar axis to the line Op
  • Polar Equation
    • r=a circle
    • α=α0 line
    • f(r,α)=0
  • Relation to Cartesian Coordinates
    • Polar to Rectangular
      • x=rcosα y=rsinα
    • Rectangular to Polar
  • Exercises
    • Transform coordinates between Cartesian Coordinates and Polar Coordinates
    • Recognize the graph of the Polar equation is a Conic/Circle/Line by changing to Cartesian Coordinates
      • 1 Change the polar coordinates to rectangular coordinates
      • 2 Replace the α in the equation
• 10.4 Parametric Representation of Curves in the Plane
  • Cycloid
    • x=a(t-sint) y=a(1-cost)
    • Theorem A
    • Exercises
      • Find the derivatives for the function determined by x=f(t), y=g(t) and determine them at a value t
        • 1 Find the derivative by using theorem A
        • 2 Substitute t by the value
      • Evaluate antiderivative includes x,y where x=f(t), y=g(t) are given
        • 1 From x=f(t), calculate dx
        • 2 Substitution x and y
        • 3 Calculate