Double Quadratic Equation

Date: 7/5/96 at 18:42:44
From: Duperray Christophe
Subject: Double Quadratic Equation

Hello,

Could you please teach an old lost student if there is a simple way to 
solve something that I would call a "double quadratic equation:"

ax^2 + bx + cy^2 + dy + e = 0

I know it's a shame, but school is so distant ...

Thanks for reply,
Greetings from southern France

J Ph Gaillard

Date: 7/5/96 at 19:23:39
From: Doctor Anthony
Subject: Re: Double Quadratic Equation

The equation you have given is the equation of a conic.  Depending on 
values of a, b etc, a conic could be a circle (a = b would be one 
condition), an ellipse, a hyperbola, a parabola or a pair of straight 
lines.  There are various conditions on the values of the coefficients 
which allow you to identify which of these it is, and I can list these 
for the general conic:

ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 should you require it.  There 
is of course no single 'solution' (it is one equation with two 
unknowns) and any point (x,y) ON the conic will satisfy the equation.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/6/96 at 8:13:47
From: Doctor Anthony
Subject: Re: Double Quadratic Equation

I replied to this question yesterday, but in case it has been lost in 
the wide blue yonder, here is what I said.

This is an equation with two variables, x and y, and as such does not 
have a 'solution' in the sense of a particular value of x or y.  In 
fact it represents a conic, and could be a circle (a=b would be one of 
the conditions for this), a parabola, an ellipse, a hyperbola or a 
pair of straight lines.  The general equation of second degree is:

ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0  and there are a variety of 
conditions on the values of a, b, h, f, g, c which determine the 
nature of the conic.  Briefly these are as follows:

(i) if a=b and h=0 the conic is a circle

(ii) if h^2<ab the conic is an ellipse with the exception of the 
special case (i).

(iii) if h^2>ab the conic is in general a hyperbola except when the 
determinant
            |a h g|
            |h b f| = 0, when it is a line pair.
            |g f c|

(iv) if a+b = 0, and in consequence h^2>ab, the conic is a rectangular 
hyperbola (or a perpendicular line pair if above determinant = 0).

(v) if h^2 = ab, the conic is in general a parabola.


Geometrically the conics are the curves of intersection of a variable 
plane and a double cone.  Each of the conics has the property that it 
is met in two points, real or imaginary, by a straight line, and 
consequently its equation must be of second degree.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/