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Eliminating ''p'', ''q'', ''r'', and λ from these equations, along with ''Xp''+''Yq''+''Zr''=0, gives the equation in ''X'', ''Y'' and ''Z'' of the dual curve.
[[File:Dual.webm|thumb|thumbtime=0.5|458px|On the left: the ellipse <math>(x/2)^2+(y/3)^2=1</math> with tangent lines <math>xX+yX=1</math> for any <math>X, Y</math>, such that <math>(2X)^2+(3Y)^2=1</math>. On the right: the dual ellipse <math>(2X)^2+(3Y)^2=1</math>. Each tangent to the first ellipse corresponds to a point on the second one (marked with the same color).]]
For example, let ''C'' be the [[Conic section|conic]] ''ax''<sup>2</sup>+''by''<sup>2</sup>+''cz''<sup>2</sup>=0. Then dual is found by eliminating ''p'', ''q'', ''r'', and λ from the equations
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