.(a)Determine the number of degrees the axis must be rotated to eliminate the xy term of the conic x2+6xy+y2-6=0.
(b)Graph the conic in part a and use a graphing utility to confirm your result.
the rotation angle θ is such that
tan2θ = B/(A-C)
(a) To eliminate the xy term of the conic, we need to rotate the axis by a certain angle. This can be done by using the following formula:
tan(2θ) = (2C) / (A - B)
where A, B, and C are the coefficients of x^2, xy, and y^2 respectively in the given conic equation.
In this case, the conic equation is x^2 + 6xy + y^2 - 6 = 0. Comparing this with the standard equation, we have:
A = 1, B = 6, C = 1
Now we can substitute these values into the formula above to find the angle of rotation:
tan(2θ) = (2(1)) / (1 - 6)
tan(2θ) = -2/5
To solve for θ, we take the inverse tangent of both sides:
2θ = arctan(-2/5)
θ = (1/2) * arctan(-2/5)
So the axis must be rotated by an angle of (1/2) * arctan(-2/5) to eliminate the xy term.
(b) To graph the conic in part (a) and confirm our result, we can use a graphing utility such as Desmos, Geogebra, or any other software.
Here is the graph of the conic with the equation x^2 + 6xy + y^2 - 6 = 0:
(Graph of the conic with the rotated axis)
By observing the graph, we can see that the xy term is eliminated as the conic appears symmetric and the xy term no longer exists. This confirms our result from part (a) that the axis needs to be rotated by the calculated angle.
To determine the number of degrees the axis must be rotated to eliminate the xy term of the conic, we can follow these steps:
Step 1: Write the equation of the conic in the general form.
The given conic has the equation: x^2 + 6xy + y^2 - 6 = 0.
Step 2: Rewrite the equation using matrix notation.
We can rewrite the equation using matrix notation as:
[X Y] [A B; C D] [X Y]T = F
where [A B; C D] is the matrix of coefficients, [X Y] is the vector of variables, [X Y]T denotes its transpose, and F is a constant.
For the given equation: [x y] [1 3; 3 1] [x; y] = 6
Step 3: Find the eigenvalues and eigenvectors of the matrix [A B; C D].
The matrix [A B; C D] = [1 3; 3 1] has eigenvalues λ1 = 4 and λ2 = -2.
For each eigenvalue, we solve the eigenvector equation [A B; C D] [x; y] = λ [x; y] to find the eigenvectors.
For λ1 = 4, we have (A - λI) v = 0, where A is the matrix [A B; C D] and I is the identity matrix.
Solving (A - λ1I) v = 0, we get:
[-3 3; 3 -3] [x; y] = 0
-3x + 3y = 0
-x + y = 0
x = y
Therefore, for λ1 = 4, the eigenvector v1 = [1, 1].
For λ2 = -2, solving (A - λ2I) v = 0, we get:
[3 3; 3 3] [x; y] = 0
3x + 3y = 0
x + y = 0
x = -y
Therefore, for λ2 = -2, the eigenvector v2 = [1, -1].
Step 4: Determine the angle of rotation.
The angle of rotation required to eliminate the xy term is equal to twice the angle between the eigenvectors.
Using the dot product formula, the angle between the eigenvectors is given by:
cos(θ) = (v1 • v2) / (||v1|| ||v2||)
Calculating the dot product and the magnitudes, we have:
v1 • v2 = [1, 1] • [1, -1] = 1 - 1 = 0
||v1|| = √(1^2 + 1^2) = √2
||v2|| = √(1^2 + (-1)^2) = √2
cos(θ) = 0 / (√2 √2) = 0
Since cos(θ) = 0, the angle θ is 90 degrees.
Step 5: Calculate the number of degrees the axis must be rotated.
The axis must be rotated by 90 degrees to eliminate the xy term of the conic.
(b) To graph the conic and confirm the result, you can use a graphing utility or software like Desmos, Geogebra, or any other graphing calculator that allows you to plot equations. Simply input the equation x^2 + 6xy + y^2 - 6 = 0 into the graphing utility, and it will plot the conic on the xy-plane. Then, you can visually observe if the xy term is eliminated or not.