Solve system of two conic equation

Question

I need to solve three types of system of equations in general form:

1. System of two linear equation ($Ax + By + C = 0$) which can be done perfectly by calculating D, Dx, Dy.

2. System of two equations which is $Ax + Bxy + Cy + D = 0$

3. System of two conic equations which is $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

My question is: is there any effective way to solve type 2 and type 3 ?

Thanks for reading. Any answers are appreciated !

Answer 1

For 2, let us consider the problem of two equations (I assume no potential accident !) such as $$A_1 x + B_1x y + C_1y + D_1 = 0$$ $$A_2 x + B_2x y + C_2y + D_2 = 0$$ From the first equation, you can eliminate $y$ and obtain $$y=-\frac{A_1 x+D_1}{B_1 x+C_1}$$ and replace $y$ in the second equation. Multiply the result by $B_1 x+C_1$ and expand. You arrive to a quadratic equation in $x$. Compute the roots for $x$ and compute $y$'s accordingly.

For 3, the problem is much more difficult and I suppose that only numerical methods could be used (Newton-Raphson for example) but they would require a reasonable starting point.

Answer 2

Hint:

For 2, for example, $$\left\{\begin{matrix}A_1x+B_1xy+C_1y+D_1=0\\A_2x+B_2xy+C_2y+D_2=0\end{matrix}\right.$$ multiply a suitable real number to one equation, and minus another to eliminate the term $xy$, then from the obtained equation, you have the relation between $x$ and $y$. the rest is for you.

For 3, it is complicate to solve this system. But we have two ways:

first, go to use the software: matlab ...

second, use the parameter equation of one equation of your system, you have $x=f(t),y=g(t)$, take them into another equation, then you have an equation of $\sin(t)$ and $\cos(t)$, and you will have an equation of $\cos(t)$(or $\sin(t)$) of order 4. then you get $t=\arccos(...)$. Examine all the solutions of $t$, you will get the solutions of you equations. Take attention of the sign of $\cos(t)$(or $\sin(t)$).