find an x^' y^'-equation for the graph of the given x y-equation so that the center or, in the c

Find an x^' y^'-equation for the graph of the given x...

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone, resulting in different types of curves like ellipses, parabolas, and hyperbolas. Identifying the type of conic is crucial to understanding its properties, such as symmetry, foci, and asymptotic behavior. The standard forms achieved through translation and completing the square help classify the conic based on the coefficients and structure of the equation.

Graphical Representation

Graphical representation involves sketching the conic along with both the original and the translated coordinate axes. This visual depiction aids in understanding the transformation effects and the geometric relationship between the original curve and its translated form, highlighting the significance of the center or vertex and other important features like axes of symmetry and asymptotes.

Completing the Square

Completing the square is an algebraic technique used to convert a quadratic expression into a perfect square plus or minus a constant. This process is essential in rewriting conic equations to reveal their key geometric features, such as the center or vertex, by allowing terms to be grouped into perfect square forms, which simplifies the identification of the conic's translation from the origin.

Translation of Axes

The translation of axes involves shifting the coordinate system so that the new origin coincides with a specific point on the conic, such as the center or vertex. This method uses the results from completing the square to determine how far and in what direction the coordinate axes need to be moved, thereby simplifying the equation of the conic and its analysis.

Transcript

00:01 In problem 40, we have this equation x squared plus 10xy plus y squared minus 6 equal to 0.

00:08 And what we are asked to do is graph this in an x prime, y prime, prime, plane, so that it doesn't have an x prime y prime term.

00:16 So that essentially just means that we have to eliminate that 10xy variable.

00:22 Now, before we begin doing that numerically, i always like to look at what's going on algebraically.

00:27 Gladly with the graphing program, we can graph that graph.

00:29 And we see that it is this rotated hyperbola here.

00:34 So we're probably going to be looking for an axis that is about right there.

00:42 And it should go through the origin, just a quick little thing here.

00:46 And this will be the theta that we're trying to find in order to convert things to x prime and y prime.

00:55 So how are we going to find that theta with this equation right here? i've already had some practice through some of the previous.

01:01 Problems for this.

01:02 So it's our a value, choose 1, minus our 1 over our b.

01:06 It's just going to be set equal to 0.

01:10 Now, what's going to be cotangent of 2 theta equal to 0? well, cotangent is cosine over sine, so our cosine has to be equal to 0, which means that, let's see, we have y, we have x, so cosine is 0 at 90 degrees.

01:29 So that's our 2 theta, so theta is, theta is 45 degrees.

01:35 Awesome.

01:37 Now that we have solved for that, we can now plug that into our equation for x and y, which we have solved plenty of times...

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