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Equation1 => $Dxy-Zx+Fx^2+Gy^2-Hy+K=0$

Equation2 => $Mxy-Nx+Ox^2+Py^2-Qy+R=0$

These equations when solve will give at most 4 points of Intersection But how to solve these two equations. The above equations when solved will give at most 4 roots

Note: here Capital Letters are all constant. I need to find the roots.

andikat dennis
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  • I use the method of "resultants", which is best done with a computer algebra system (Maple, Mathematica) because the intermediate expressions can get pretty involved (especially when using symbolic constants). – Blue Jun 20 '13 at 13:59
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    A "by hand" approach would be something like this: Eliminate $y^2$ from the equations (compute $P E_1 - G E_2)$. This gives you a polynomial equation in which the highest power of $y$ is $1$; solve that polynomial for $y$, and substitute the solution into either original equation (note that you'll be squaring a large polynomial as part of the process), and you'll have a quartic polynomial in $x$. From there ... well ... quartics aren't easy to solve symbolically ---check the web--- but, once you have your $x$ values, you can substitute them into the earlier $y$ formula, and you're done! – Blue Jun 20 '13 at 14:04
  • This may also help you get started: http://en.wikipedia.org/wiki/Conic_section#Intersecting_two_conics – cobaltduck Jun 20 '13 at 14:35
  • This is a spin-off from this question. Slightly different formulation for the conics, but essentially the same question, so you might even call it a duplicate. Note that you can solve this problem using only cubic equations, not quartic ones. See my answer for details. – MvG Jun 20 '13 at 21:53

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