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

Rotation of Axes

This concept involves changing the coordinate system by rotating it to simplify the equation of a conic section. By applying a coordinate transformation, one can eliminate the cross-product xy-term, making it easier to classify and analyze the conic. The transformation uses trigonometric formulas to relate the old coordinates to the new ones, ensuring that the conic's shape and properties remain invariant under rotation.

Elimination of the xy-Term

Removing the xy-term from a quadratic equation simplifies the expression by decoupling the variables. This is accomplished by selecting an appropriate rotation angle that transforms the equation into one without products of the new variables. This process not only aids in identifying the type of conic section but also makes further analysis, such as completing the square or graphing, more straightforward.

Determination of the Rotation Angle

The rotation angle necessary to remove the xy-term is found by using the formula tan(2?) = B/(A?C), where A, B, and C are the coefficients of x², xy, and y², respectively. This formula arises from the conditions imposed by the rotational transformation and provides an exact measure of how much the original axes must be rotated to achieve a simplified equation.

Conic Section Classification

After eliminating the xy-term, the resulting equation is easier to classify into one of the standard conic sections: ellipse, parabola, or hyperbola. This classification is determined by examining the coefficients of the squared terms in the transformed equation. Understanding the type of conic is essential for further analysis, such as determining its center, vertices, and asymptotes, and for accurately graphing the conic section.

Transcript

00:01 We are given an equation and we are asked to rotate the axes so that the new equation will contain no xy term.

00:09 And then to analyze and graph this new equation.

00:15 The equation we're given is 11x squared plus 10 root 3xy plus y squared minus 4 equals 0.

00:23 From a previous problem, we determined that the appropriate angle of rotation theta was 30 degrees, and that the rotation formula were x equals cosine of 30 degrees.

00:40 So one -half times root 3 x -prime minus y -prime, and y equals one -half times x -prime plus root 3 y -prime.

01:01 Substituting x and y in terms of x prime and y prime we get 11 times one half squared so 11 over 4 times and foiling we get 3 x prime squared minus 2 root 3 x prime y prime plus y prime plus the next turn so this is 10 root 3 x prime y prime plus y prime squared plus the next turn so this is 10 root 3 times 1 4th, so this is 5 root 3 over 2 times and then 3x prime minus y prime times x prime plus root 3 y prime.

01:54 This gives us root 3 x prime squared plus 3 x prime y prime minus x prime y prime is plus 2 x prime y prime minus root 3 y prime squared.

02:19 And plus y squared, which is plus one -fourth times x -prime squared, plus 2 root 3, x -prime, y -prime, plus 3 y -prime, plus 3 y -prime squared.

02:39 And then moving the constant to the other side, we get equals 4, multiplying both sides by 4, and also adding like terms, we get 11 times 3 is 33, plus 5 times root 3 over 2 times 4 is 10 times root 3 times root 3, 10 times 3 is 30 plus 1.

03:19 This is 31 plus 33 or 64 x prime squared.

03:33 We have the mixed term is going to cancel out.

03:36 And we have plus 11 plus 10 root 3.

03:47 Minus minus negative root 3 is minus 10 times 3 or minus 30 and then plus 3 this is 14 minus 30 or negative 16 so this is minus 16 y prime squared equals 16 and dividing by 16 you put this into standard form so that we get x prime squared over and then 64 divide by 16 is 4, so x prime squared over 1 fourth minus y prime squared equals 1.

04:45 And just by looking at our formula, our new equation, we see that this is going to be a hyperbola in the x prime, y prime plane.

04:58 To analyze this hyperbola, notice that it has a center 0 -0.

05:04 We see that the coefficient of x is positive.

05:21 So, a squared equals one -fourth, and b -squared equals 1.

05:29 So we have that c -squared, which for our hyperbola is equal to a -square plus b -square.

05:41 It's 1 to 5 -4ths.

05:44 So that c will be equal to five -halfs.

05:56 And so we have that transverse axis is going to be.

06:00 Parallel to the x -axis.

06:04 In fact, the transverse axis is going to be the x -prime axis itself.

06:17 We also have that the gauci will be located at plus or minus c -0, so plus or minus root 5 over 2, 0, in the x -prime y prime plane...