Proof Prove that the graph of the equation A x^2+C y^2+D x+E y+F=0 is one of the following (exce

Proof Prove that the graph of the equation A x^2+C y^2+D x+E...

Key Concepts

Degenerate Cases

Degenerate cases occur when the quadratic equation does not represent a typical conic section but rather a simpler geometric entity such as a point, a line, or a pair of intersecting lines. These cases arise when the equation’s parameters satisfy specific conditions that collapse the conic into a lower-dimensional figure. Recognizing and handling these cases is important in a complete analysis of conic sections.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. This family of curves comprises circles, ellipses, parabolas, and hyperbolas. Their properties and shapes depend on the angle at which the plane intersects the cone, and they are fundamental objects in both algebra and geometry, illustrating the deep connection between algebraic equations and geometric figures.

General Quadratic Equations in Two Variables

The general quadratic equation in two variables, Ax² + C y² + Dx + Ey + F = 0, provides a unified algebraic framework for representing conic sections. By analyzing the coefficients of the quadratic and linear terms, one can determine the type of conic section described by the equation. This equation serves as the starting point for classifying the curve based on its algebraic structure.

Completing the Square

Completing the square is an algebraic technique used to rewrite quadratic expressions in a perfect square form. In the context of conic sections, this method is essential for transforming the general quadratic equation into its standard form, which clearly exposes the center, vertices, foci, and other geometric features of the curve. This standardization is crucial for identifying and analyzing the specific type of conic section represented by the equation.

Classification via Coefficient Conditions

The classification of conic sections from the general quadratic equation depends on the relationships among the coefficients. For example, when the coefficients of x² and y² are equal (A = C), the conic is a circle, whereas if one of these coefficients is zero (while the other is nonzero), the graph represents a parabola. Additionally, if both A and C are nonzero and of the same sign (AC > 0), the conic is an ellipse (or a circle as a special case), and if A and C have opposite signs (AC < 0), the graph is a hyperbola. These conditions help in determining the exact nature of the curve without needing to graph it directly.

Transcript

00:01 The first condition, a equals c, we have to prove that when this is true, this equation, ax squared plus cx squared plus dx plus ey plus s equals 0, this equation of a circle.

00:16 And i'm just going to write a for c, so it'll be ax squared plus ay squared plus dx plus ey, once s equals 0.

00:34 Now this will be the equation of a circle.

00:36 If we can get it into standard form for a circle, so of the form x minus h squared, plus y minus k squared, equals r squared.

00:45 So we can do that by completing the square.

00:47 And before completing the square, we need to make these coefficients a 1, so let's divide everything by a.

00:54 So if we divide this first term by a, we get a squared.

00:57 Divide a second term by a, we get y squared.

01:01 Now the next coefficients will change a little bit.

01:02 We're going to have b over a.

01:04 D over a x plus e over a y plus f over a and 0 divided by a 0 on the right side and now we can complete the square let's complete the square in the x terms so x squared plus d over a x and then plus term we're going to add is half of this squared so it will be d over 2a squared and now similar is going to be the exact same for the y actually so we'll have y squared plus e over a y or very similar rather plus e half of e over a a we're adding that squared and let's move the f over a to the other side of the and since we're adding this term on the left side we also need to add it to the right side so let's add d over 2a squared plus t over 2a squared so now the left side we can factor each of b so with the x is going to make x plus d over 2a squared yeah this is x plus d over 2a squared yeah this is x plus d over 2a squared and then that's x term the y term will be y plus factoring this e over 2a squared and in the right side we're going to have all these terms here negative for a plus d over 2a squared plus e over 2a squared now this equation is the equation of a circle since this is x minus h plus y minus k squared and then r squared in this side a constant on this side this is the equation of a circle so you can do a similar thing in the next condition next condition says that if a equal zero or c equals zero and we should get the equation of a parabola excuse me a parabola the equation of a parabola and i'm just going to do the case for c equals zero so we're going to do c equal 0 and the case with a equal 0 is exactly the same, just switch x and y.

04:17 So when p equal 0, all we have is an a x squared.

04:25 There's no y squared anymore, you can c is 0, but we're going to have plus bx, plus f equals 0, and again we can complete the square on the x term before doing that, or let's actually just factor out in a, it'll be a easier.

04:47 This time we're going to factor out an a.

04:50 X squared, factoring an a out of this term makes d over a.

04:58 X, and then the number we need to add again is d over 2a.

05:06 All right.

05:08 And then plus ey plus f.

05:14 Now on the left side of the equation, we've added, we've actually added d over a.

05:22 Excuse me, we need to add this squared.

05:25 Sorry, we need to just adding this thing.

05:28 Squared, right, i was on this.

05:32 I think should be squared here.

05:35 D over 2a squared, that's the term we're adding.

05:40 And if we're adding d over 2a squared, and over here, what are we adding? we're adding d over 2a squared times a.

05:49 So we're actually adding d, i just write it on like that, a times d over 2a squared.

06:06 That's an a.

06:06 Alright, and then next, you can factor the left side.

06:11 X is an x plus d over 2a all squared and on the right side and let's move all these terms to the right side so we're going to get minus ey minus f plus a times d over 2a squared all right next up let's divide each side by a so on the left side we're just going to have x plus d over 2a squared squared.

06:55 That's exactly what we want for the standard form of a parabola.

07:00 On the right side we're going to have negative e over a.

07:07 We've previously had negative e.

07:09 And negative e over a, y minus f over a.

07:15 Now dividing this term by a just cancels out that a in front.

07:21 So plus c over to a squared.

07:26 All right.

07:29 And finally, what we can do is you can factor out an e over a in this side.

07:37 And we'll have y with some other term here.

07:41 And therefore, this is the standard form of a parabola, which opens upward.

07:45 We have x minus h squared.

07:46 And you know, this will be equal to 4p, some constant, and the y minus k.

07:52 Okay.

07:53 So next we had the condition that ac is positive, so greater than zero.

08:00 And it should be the equation of an ellipse.

08:03 So in this case, we have all of the constants.

08:08 And we're essentially going to do the same thing that we did for number one, except that's going to be a different variable now.

08:15 That's going to still be a c, excuse me, a different concept.

08:19 So c y squared, plus eo d x plus e1, plus f equals 0.

08:36 All right.

08:37 And again, we need to complete the same.

08:38 Square.

08:39 I'm going to do that, i'm going to factor an a out of the x terms.

08:43 So we'll have x squared, plus factoring a out of here makes a d over a.

08:48 You've seen this before.

08:51 So just a class time, we're going to add, oh, sorry, that should be d over an a.

08:59 D over a, x, plus, and then we're adding half of that square, complete the square, b over 2a squared, plus, and now the y term, factor out of c, or y squared...

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