Use traces to sketch and identify the surface. 36 x^2+y^2+36...

Key Concepts

Ellipsoids

An ellipsoid is a type of quadric surface described by an equation of the form x²/a² + y²/b² + z²/c² = 1, where a, b, and c represent the lengths of the semi-axes along the x, y, and z axes respectively. This surface is a three-dimensional analogue of an ellipse and is closed and bounded, meaning it is finite in extent.

Traces

Traces refer to the curves obtained by intersecting a three-dimensional surface with coordinate planes, or more generally, planes parallel to these axes. By examining the shape of these intersections (such as circles or ellipses), one can gain insight into the overall geometry of the surface and aid in its visualization.

Standard Forms of Quadric Surfaces

Converting a quadratic equation into its standard form is key to identifying the type of surface it represents. This involves algebraic manipulation, such as dividing by a constant or completing the square, to rewrite the equation in a form that clearly shows the relationship between the squared terms. Recognizing the standard form allows for the classification of the surface (e.g., ellipsoid, hyperboloid, paraboloid) and provides information about its dimensions and orientations.

Transcript

00:01 Alright, so in this video we're given 36x squared plus y squared plus 36 c squared is equal to 36, and we're asked to use traces in order to identify the surface and grab it.

00:17 So the technique is we're going to set the variables equal to k and see what we get.

00:25 So if we're at x equal to k, we're going to get 36k squared plus y squared, plus 36 z squared is equal to 36.

00:35 Moving the constants to the other side, we get y squared plus 36 z squared is equal to 36 minus 36k squared.

00:45 And now if we want to make the right -hand side equal to 1, we're going to divide by 36 minus 36k squared.

00:53 So we get y squared divided by 36 minus 36k squared, plus 36 z squared, divided by 36 minus 36 k squared is equal to 1.

01:06 And now we can rewrite this because, so this looks really similar to an ellipse, but remember for an ellipse, we're gonna need y squared divided by some constant squared plus z squared divided by some constant squared has to equal to 1.

01:23 So what we can do is we can write 36 minus 36k squared as the square root of 36 minus 36k squared squared squared.

01:33 So we can write it as a square root of 36 minus 36k squared squared.

01:38 So now rewriting everything via y squared divided by the square root of 36 minus 36k squared squared plus z squared divided by the square root of 36 minus 36k squared divided by 6 squared is equal to 1.

01:54 And immediately we recognize this as an ellipse.

01:57 Now do we have any constraints on k? well, what we know is a denominator cannot be zero, and whatever is under the square root cannot be negative.

02:06 So we have 36 minus 36 k squared is greater than 0.

02:11 Rearranging, we get 36 is greater than 36 k squared.

02:17 So k squared has to be less than 1.

02:20 So that means that we have a constraint on our x, in that it has to be between negative 1.

02:31 Okay, now we're going to set y equal to k.

02:35 So we plug that into the surface that we're given, and then moving the constants to one side, we get to the right side, we get 36x squared plus 36 z squared is equal to 36 minus k squared, dividing by 36 minus k squared because we want our right -hand side to be equal to 1.

02:54 We get 36 x squared divided by 36 minus k squared plus 36 z squared divided by 306 z squared, minus k squared is equal to 1.

03:03 So this right over here.

03:06 And now we can rewrite this as x squared divided by the square root of 36 minus k squared divided by 6, all of it squared plus z squared divided by the square root of 36 minus k squared divided by 6 squared is equal to 1...

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