Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation

Rotate the axes so that the new equation contains no...

Key Concepts

Classification of Conic Sections

Classification of conic sections involves identifying the type of conic (ellipse, hyperbola, or parabola) by examining the quadratic coefficients and the discriminant. Once the cross term is eliminated, the standard form of the conic can be used to determine its geometric characteristics.

Elimination of the Cross Product Term

Elimination of the cross product (xy) term is a key process in simplifying the general quadratic form. By rotating the axes, the term that couples x and y is removed, yielding an equation in standard form where the conic's properties, such as orientation and type, become more apparent.

Graphing Conic Sections

Graphing conic sections requires understanding the conic's key features like its center, vertices, axes, and, if applicable, asymptotes. After simplifying a conic equation via rotation and completing the square, these features are used to accurately sketch the conic's graph and visualize its orientation and dimensions.

Rotation of Axes

Rotation of axes is a technique used to eliminate the cross (xy) term in a quadratic equation by transforming the coordinate system. It involves applying a rotation transformation with a specific angle, derived from the coefficients of the original equation. This simplification makes the equation easier to analyze and classify.

Coordinate Transformation

Coordinate transformation involves changing from one coordinate system to another to simplify the mathematical expression or to align the analysis with the geometry of the problem. In conic sections, such a transformation can eliminate mixed terms and center the conic, facilitating further analysis.

Transcript

00:01 We are given an equation and we are asked to rotate the axes so that the new equation will contain no mixed term and then to analyze and graph this new equation.

00:13 The equation is 4x squared minus 4xy plus y squared minus 8 root 5x minus 16 root 5y equals 0.

00:25 Recall from a previous problem, we found that the angle of rotation is theta equals 90 degrees minus 1⁄2, inverse, co2.

00:33 Tangent three -fourths which is approximately 36 .87 degrees and we also found that the rotation formulas are x equals 1 over root 5 x prime minus 2 y prime and y equals 1 of a root 5 to x prime plus y prime substituting in x and y for x prime and y prime so we're gonna change variables here and we get 4 times 1 fifths, so we get 4 fifths, and foiling, we get x prime squared, minus 4 x prime, y prime, plus 4 y prime squared.

01:51 The next term is minus 4 xy, so this is minus 4xy, so this is minus 4 times 1 fifth, or minus four fifths, and then foiling we get 2x prime squared, plus x prime, x prime y minus 4 x prime y prime so minus 3 x prime y prime plus or minus 2 y squared plus y squared so plus 1 5th 4 x prime squared plus 4 x prime y prime plus y prime plus y prime squared minus 8 root 5 times x which is minus 8 fifths times x prime minus 2 y prime minus 16 root 5 times y which is minus this is simply minus 8th and 8th and then this is going to be minus 16 times 2x prime plus y prime equals 0 and multiplying both sides by 5 and canceling out terms we get 4 x -prime squared minus 8 x -prime squared plus 4 x -prime squared this is going to be 8 minus 8 is 0 so 0 x -prime squared the mixed term x -prime y -prime disappears and then for y prime squared we have 4 times 4 is 16 y prime squared minus four times negative to a plus 8 y prime squared and plus 1 y prime squared this is going to be 25 y prime squared for the x term or x prime term we have plus negative 8 x prime minus 32 x prime which is going to be so minus 40 x prime and this actually is not true this is going to be 5 times 40 so 200 negative 200 x prime and then we have negative 40 times negative 2 is positive 80 y prime and and then minus 16 times 5 is minus 50 plus 30 is 80.

06:21 And so we have no y prime term.

06:29 And this is simply equal to zero.

06:32 And so we can rewrite this in the standard form as y prime squared equals 200x prime over 25, which is 8x prime.

07:03 This is our new equation.

07:05 And we can see just by looking, this is one to be a parabola in the x prime, y prime plane.

07:19 To analyze the parabola, notice that we have a vertex at zero zero.

07:52 Notice that, so we have that our equation has the form y squared equals four times two times x.

08:04 So this is 5 prime squared equals 4 times 2 times x prime, or 2 is greater than 0.

08:20 And so we have that the focus is going to be a0, where a is equal to 2.

08:33 So the focus adds 2 0 in the x prime, y prime plane.

08:41 We have a directrix with equation x equals negative a, x prime equals negative a.

09:03 So x prime equals negative two.

09:09 And we have an axis of symmetry, which is parallel to the x prime axis.

09:25 In fact, the axis of symmetry is the x prime axis.

09:30 And we also have that this parabola is going to open toward the right.

09:43 The orientation is opens right...

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