Mathematics Dictionary
Dr. K. G. Shih

    Conic Sections


  • Q01 | - Circle
  • Q02 | - Ellipse
  • Q03 | - Hyperbola
  • Q04 | - Parabola
  • Q05 | - Elliminate x*y terms

    Q01. Circle
    • Keyword | Circle equations in various coordinates
      • Equations of circle in rectangular form
      • Equations of circle in polar form
      • Equations of circle in parametric equations
    • Keyword | Circle : Draw 4 circles to tangent triangle ABC
      • 1. Draw 1 in-circle of triangle ABC
      • 2. Draw 3 ex-cirlce of triangle ABC

    Go to Begin

    Q02. Ellipse

    Ellipse
    • Subject | Ellipse : Convert (x/a)^2 + (y/b)^2 = 1 to polar form
    • Subject | Ellipse : Polar form
      • Convert R = (D*e)/(1 - e*cos(A)) to rectangular form
      • Convert R = 1.8/(1 - 0.8*cos(A)) to rectangular form
    • Keyword | Ellipse : Sketch using ruler
    • Keyword | Ellipse : Sketch using string
    • Keyword | Ellipse : Sketch tangent using reflection

    Go to Begin

    Q03. Hyperbola

    Hyperbola
    • Keyword | Hyperbola : Formula and examples
    • Keyword | Hyperbola : Locus
      • 1. Two fixed points F and G. If PF - PG = 8, Find locus of P
      • 2. Prove locus of hyperbola is (x/a)^2 - (y/b)2 = 1
    • Keyword | Hyperbola : (x/a)^2 - (y/b)^2 = -1
      • 1. Comparison : (x/a)^2 - (y/b)^2 = -1 and (x/a)^2 - (y/b)^2 = 1
      • 2. Prove that x = sec(t) and y = tan(t) is a hyperbola
    • Keyword | Hyperbola : R = (D*e)/(1 - e*cos(A))
      • 1. Convert (x/4)^2 - (y/3)^2 = 1 to polar form
      • 2. Convert R = 2.25/(1 - 1.25*cos(A)) to rectangular form
    • Keyword | Hyperbola : R = (D*e)/(1 + e*cos(A))
      • 1. Compare R = (D*e)/(1 - e*cos(A)) and R = (D*e)/(1 + e*cos(A))
      • 2. Study hyperbola x*y = 1
    • Keyword | Hyperbola : x*y = 1
      • 1. Compare x*y = 1 and (x/a)^2 - (y/b)^2 = 1
      • 2. Find focal length and equation of directrix of hyperbola x*y = 1
    • Keyword | Hyperbola : Sketch tangent using reflection

    Go to Begin

    Q04. Parabola

    Parabola
    • Keyword | Parabola : Defintion and Equations
    • Keyword | Parabola : Locus y = (x^2)/(2*D) - D/2
    • Keyword | Parabola : Sketch
      • Convert R = D/(1 - sin(A)) to rectangular form
    • Keyword | Parabola : Polar form
      • 1. Compare R = D/(1 - sin(A)) with R = D/(1 - cos(A))
      • 2. Compare R = D/(1 - sin(A)) with y = (x^2)/(2*D) - D/2
      • 3. Compare R = D/(1 - sin(A)) with y = a*(x^2) + B*x + c
    • Keyword | Parabola : Quadratic function
      • 1. Properties of y = a*x^2 + b*x + c
      • 2. Parabola of y = a*x^2 + b*x + c
    • Keyword | Parabola : Sketch using ruler
    • Keyword | Parabola : Sketch tangent using reflection

    Go to Begin

    Q05. Elliminate x*y terms

    Equation in implicity form
    • Equation : F(x,y) = A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0.
    Rotation : Transform x and y to u and v
    • (1) x = u*cos(t) - v*sin(t)
    • (2) y = u*sin(t) + v*cos(t)
    Equation after rotation : A'*u^2 + B'*u*v + C'*v^2 + D'*u + E'*v + F' = 0
    • A' = A*cos(t)^2 + B*cos(t)*sin(t) - C*sin(t)^2
    • B' = (C - A)*sin(2*t) + B*cos(2*t)
    • C' = A*sin(t)^2 - B*cos(t)*sin(t) + C*cos(t)^2
    • D' = D*cos(t) + E*sin(t)
    • E' = -D*sin(t) + E*cos(t)
    • F' = F
    Eliminate u*v term :
    • Let B'=0
    • (C - A)*sin(2*t) + B*cos(2*t) = 0
    • Hence tan(2*t) = B/(A - C)
    • Angle 2*t = arctan(B/(A - C))
    Chnage A'*u^2 + C'*v^2 + D'*u + E'*v + F' = 0 to ((u-h)/a)^2 + ((v-k)/b)^2 = 1
    • Since there is no u*v, we can use completing square nethod
    • Hence we find h and k
    • We can also find a and b
    • We can also find vertice and focal length f
    Example : Change x*y = 1 to standard form
    Go to Begin

    Q06. Elliminate x*y terms

    Equation in implicity form
    • Equation : F(x,y) = A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0
    Rotation : Transform x and y to u and v
    • (1) x = u*cos(t) - v*sin(t)
    • (2) y = u*sin(t) + v*cos(t)
    Substitute x and y into F(x,y)
    • A*(u*cos(t) - v*sin(t))^2
    • = A*(u^2)*cos(t)^2 - 2*A*u*v*cos(t)*sin(t) + A*(v^2)*sin(t)^2
    • B*(u*cos(t) - v*sin(t))*(u*sin(t) + v*cos(t))
    • = B*(u^2)*cos(t)*sin(t) + B*u*v*(cos(t)^2)
    • - B*u*v*(sin(t)^2) - B*(v^2)*cos(t)*sin(t)
    • C*(u*sin(t) + v*cos(t))^2
    • = C*(u^2)*sin(t)^2 + 2*C*u*v*cos(t)*sin(t) + C*(v^2)*cos(t)^2
    • D*(u*cos(t) - v*sin(t)) = D*u*cos(t) - D*v*sin(t)
    • E*(u*sin(t) + v*cos(t)) = E*u*sin(t) + E*v*sin(t)
    • F = F
    • New coefficients : A', B', C', D', E', F'
    • A'*u^2 + B'*u*v + C'*v^2 + D'*u + E'*v + F' = 0
    Equation after rotation : A'*u^2 + B'*u*v + C'*v^2 + D'*u + E'*v + F' = 0
    • A' = A*cos(t)^2 + B*cos(t)*sin(t) - C*sin(t)^2
    • B' = (C - A)*sin(2*t) + B*cos(2*t)
    • C' = A*sin(t)^2 - B*cos(t)*sin(t) + C*cos(t)^2
    • D' = D*cos(t) + E*sin(t)
    • E' = -D*sin(t) + E*cos(t)
    • F' = F

    Go to Begin
    • Show Room of MD2002 Contact Dr. Shih Math Examples Room
    Copyright © Dr. K. G. Shih. Nova Scotia, Canada.
    1