Double Quadratic EquationDate: 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/ |
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