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Mathematics Dictionary
Dr. K. G. Shih
Elliminate x*y terms in F(x,y)
Subjects


  • AN 17 00 | - Outlines
  • AN 17 01 | - Conic section : F(x,y)
  • AN 17 02 | - Elliminate x*y term
  • AN 17 Q03 | - Change x*y = 1 to (u/a)^2 - (v/b)^2
  • AN 17 04 | -
  • AN 17 05 | -
  • AL 17 06 | -
  • AN 17 07 | -
  • AN 17 08 | -
  • AN 17 09 | -
  • AN 17 10 | -
  • AN 17 00 | - Outlines
    • Answers


    AN 17 01. Conic section : F(x,y) = 0

    F(x,y) = A*x^2 + B*x*y + C*y^2 + D*x + e*y + F = 0
    • Discriminant Q = B^2 - 4*A*C
      • Q EQ 0, it is parabola
      • Q LT 0, it is ellipse
      • Q GT 0, it is hyperbola
    • It is a circle if A EQ C and B EQ 0
    How to sketch F(x,y) on computer ?
    • Method 1 : Express y = G(x)
    • Method 2 : Elliminate x*y term by rotating an angle U
      • Then we can change it as standard form ((x-h))^2 + ((y-k)/b)^2 = 1
      • Then we can plot the graph in Oxy system
      • Then rotate the plot by an angle of A

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    AN 17 02. Elliminate x*y term

    F(x,y) = A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0
    • Question
      • Eliminate x*y term by rotating an angle
      • The angle is tan(Ang) = (A - C)/B
    • After rotation : F(,u,v) = P*u^2 + Q*u*v + R*v^2 +S*u + T*v + W = 0
      • Rotation : u = x*cos(Ang) + y*sin(Ang) and v = -x*sin(Ang) + y*cos(Ang)
      • Substitute rotation formula into F(x,y) = 0 and simplify we have
        • P = A * Cos(ANG) ^ 2 + C * Sin(ANG) ^ 2 + B * Cos(ANG) * Sin(ANG)
        • Q = 2*(C - A)*cos(Ang)*sin(Ang) + B*(cos(Ang)^2 - sin(ANg)^2)
        • R = A * Sin(ANG) ^ 2 + C * Cos(ANG) ^ 2 - B * Cos(ANG) * Sin(ANG)
        • S = D * Cos(ANG) + E * Sin(ANG)
        • T = -D * Sin(ANG) + E * Cos(ANG)
        • W = F
    • To elliminate x*y term means Q = 0 in F(u, v)
      • Hence 2*(A - C)*cos(Ang)*sin(Ang) = B*(cos(Ang)^2 - sin(Ang)^2)
      • Hence (A - C)*sin(2*Ang) = B*cos(2*Ang)
      • Hence tan(2*Ang) = (B)/(A - C)
    • or Ang = 0.5*arctan(B/(A - C))

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    AN 17 03. Change x*y = 1 to (u/a)^2 - (v/b)^2

    Example : Rotate 45 degrees of x*y = 1
    • A = 0, B = 1, C = 0, D = 0, E = 0, F = -1
    • Hence tan(2*Ang) = (B)/(A - C) = infinite and Ang = pi/2
    • Hence Ang = 45 degrees
    • P = +B*cos(ang)*sin(Ang) = 1/2
    • R = -B*cos(ANg)*sin(Ang) = -1/2
    • S = 0 and T = 0
    • W = -1
    • After rotating 45 degree, the new equation (u^2)/2 - (v^2)/2 - 1 = 0
    • Standard hyperbola form is (u/Sqr(2))^2 - (v/Sqr(2))^2 = 1
    Computer method
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    AN 17 04.


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    AN 17 05.


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    AN 17 06. New


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    AN 17 07. Answer

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    AN 17 08. Answer

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    AN 17 09. Answer

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    AN 17 10. Answer
    • Ang = 0.5*arctan(B/(A - C))

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    AN 17 00. Outlines

    Elliminate x*y terms in F(x,y)
    • Rotate angle : Ang = 0.5*arctan(B/(A - C))

    Go to Begin
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