Graph the equation relative to the x^' y^' system in which the equation has no x^' y^' term. x^2

Graph the equation relative to the x^' y^' system in which...

Key Concepts

Diagonalization of the Quadratic Form

Diagonalization of the quadratic form refers to the process of converting the quadratic part of a second-degree equation into a form that has no cross-product term. This is typically achieved by applying a suitable rotation to the axes, which simplifies the conic section's equation and aids in its classification and graphical representation.

Rotation of Axes

Rotation of axes is a technique used to eliminate the mixed term (such as xy) in a quadratic equation by transforming the coordinate system through a rotation. This process simplifies the equation by aligning the axes with the principal directions of the conic, which makes it easier to classify and graph the conic section.

Conic Sections

Conic sections are curves that arise from the intersection of a plane with a double-napped cone, and they are generally represented by second-degree equations. The presence of mixed terms indicates that the conic may be rotated relative to the coordinate axes, and transforming the equation can reveal whether the conic is an ellipse, parabola, or hyperbola.

Coordinate Transformation

Coordinate transformation involves changing the coordinate system, such as from (x, y) to (x', y'), through methods like rotation to simplify the equation of a curve. This is critical for eliminating cross terms and rewriting the equation in a standard form that is easier to analyze and graph.

Transcript

00:01 All right, so we're going to rotate, find out how many degrees this has been rotated, write the equation of the rotated conic, and then do a quick sketch.

00:14 All right, so here's our equation.

00:18 We use the cotangent formula, and we've got a minus c equals zero.

00:23 So that means that contention of two theta is zero.

00:26 Okay, well, that means that theta has to be 90 degrees because the adjacent side is zero.

00:35 I mean, 2 theta is 90 degrees, which means theta equals 45 degrees.

00:42 Okay, so now we know that whatever this graph is, whatever this graph ends up being, it's going to be, we're going to graph it on a 45 degree rotation.

00:58 And graph in on these axes here.

01:03 All right, so we're pretty familiar with this, with 45 degrees, so with 45 degrees, x is equal to x prime minus y prime over the square root of two, and y is equal to x prime plus y prime over the square root to two.

01:28 You plug those into our original function, and we get x prime.

01:37 Minus y prime over two squared plus 10 x prime minus y prime over the square root of two that's the x times the y which is x prime plus y prime over square root of two and then our last term well the second last we have a constant is going to be x prime plus y prime over the square root of two squared and and that's minus six equals zero.

02:14 Okay, now let's go ahead and square this.

02:17 So this is gonna be x prime squared minus two, x prime y prime plus y prime squared over two plus ten times x prime squared...

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