Parabola definition parboiling my frontal lobe

[ngm01]

Parabola definition parboiling my frontal lobe!

Homework Statement

The general equation of second degree is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. the values of the coefficients in the general eqn. for a parabola shown above could be:

A B C D E F
+ve 0 0 0 +ve 0
-ve 0 0 0 +ve 0
0 0 +ve +ve 0 0
0 0 +ve -ve 0 0 *** This is the combination that works?
0 -ve +ve 0 0 0

Homework Equations

Conic section equation where from the general equation:
B^2 - AC = 0 (Parabola)
B^2-AC < 0 (Ellipse),
B^2-AC > 0 (Hyperbola),
A=C and B=0 (Circle)
A=B=C=0 straight line

The Attempt at a Solution

I started by using the conic section coefficient guide for a parabola and substituted the possible values of the coefficients given in the table above into the conic section equation to try to eliminate a combination that would be a ellipse or hyperbola and the infer by the process of elimination that..." Aha these coefficients can define a hyperbola so it can't work!..." but they all satisfy pretty much any of the guides --so that went down in flames, I really don't see what makes that combination of coefficients special for a parabola. So if you do please help me understand.

Regards

 

[mighty2000]

,
Frustrated Scientist

Dear Frustrated Scientist,

I can understand your frustration with trying to understand the definition of a parabola using the general equation of second degree. However, the equation you have provided is not the standard form of a parabola. The standard form for a parabola is y = ax^2 + bx + c, where a, b, and c are constants.

To better understand the definition of a parabola, we must first understand its shape. A parabola is a symmetrical curve that resembles an arch. It has a vertex, which is the highest or lowest point of the curve, and it extends indefinitely in both directions. The general equation of a parabola in standard form can be rewritten as y = a(x - h)^2 + k, where (h, k) is the coordinates of the vertex.

Now, let's look at the general equation of second degree that you provided. In order for it to represent a parabola, the coefficients A, B, and C must satisfy the condition B^2 - 4AC = 0. This is known as the discriminant, and it determines the type of conic section the equation represents. When B^2 - 4AC = 0, the graph of the equation will be a parabola. This is because the discriminant is equal to the constant term in the standard form of the parabola, c.

In the table provided, the only combination that satisfies the condition B^2 - 4AC = 0 is when A = 0, B = -ve, and C = +ve. This means that the equation will have the form y = bx^2 + c, where b is a negative constant and c is a positive constant. This is the standard form of a parabola with its vertex at the origin.

I hope this helps clarify the definition of a parabola and how to determine its form using the general equation of second degree.A fellow scientist