(a)Determine the number of degrees the axis must be rotated to eliminate the xy term of the conic x2+6xy+y2-6=0.
tan2θ = B/(A-C)
how do I answer the question?
Huh? Use your numbers:
cot2? = (1-1)/6 = 0
2? = 90°
? = 45°
http://www.wolframalpha.com/input/?i=x%5E2%2B6xy%2By%5E2-6%3D0
Well, aren't you conic shape-tastic! To eliminate that pesky xy term, we need to rotate the axis. So grab that protractor, my friend, because we're about to do some conic cartwheels!
First things first, we need to figure out the values for A, B, and C in your conic equation, which are the coefficients of x², xy, and y², respectively. In this case, A = 1, B = 6, and C = 1.
Now, let's plug those values into the formula tan(2θ) = B/(A-C). Substituting the values, we get tan(2θ) = 6/(1-1). But uh-oh, that denominator is zero! That's a no-go in Clownville. So, we can't directly calculate the angle θ using this formula since it's undefined.
The xy term isn't ready to take a hike just yet. But hey, cheer up! I've got another trick for you. We can use a different formula to solve this problem. Let's calculate θ using tan(θ) = B/(A-C). Plugging in our values, we get tan(θ) = 6/(1-1), which simplifies to tan(θ) = 6/0. Here we go again, that nasty zero in the denominator.
Sadly, no amount of Clown magic can make this calculation work. It looks like this conic equation is resistant to being axis-rotated to eliminate that xy term. Time to get out the conic termite spray or consult with a math pro for some extra help!
To determine the number of degrees the axis must be rotated to eliminate the xy term of the conic x^2 + 6xy + y^2 - 6 = 0, you can follow these steps:
Step 1: Identify the coefficients A, B, and C in the equation of the conic. In this case, A = 1, B = 6, and C = 1.
Step 2: Substitute the values of A, B, and C into the equation tan^2θ = B/(A-C).
tan^2θ = 6/(1-1)
= 6/0
Since division by zero is undefined, we cannot directly use this formula to find the value of θ.
However, we can still determine the angle by taking the limit as C approaches A. Let's do that in the next steps.
Step 3: Calculate the limit as C approaches A.
lim(C -> A) (6/(A-C))
= lim(C -> A) (6/(1-C))
We can calculate this limit:
lim(C -> 1) (6/(1-C))
= 6/(1-1)
= 6/0
Again, we encounter division by zero, which is undefined. Therefore, the equation does not have a simple solution for the angle of rotation that eliminates the xy term.
In this case, it suggests that either the conic cannot be rotated to eliminate the xy term, or there may be another approach or method to solve the equation.
Please note that division by zero is undefined in mathematics, and further analysis or alternative techniques may be required in cases like this.
To determine the number of degrees the axis must be rotated to eliminate the xy term of the conic x^2+6xy+y^2-6=0, you can follow these steps:
1. Write the equation of the conic in the general form: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. In this case, A = 1, B = 6, C = 1, D = 0, E = 0, and F = -6.
2. Calculate the value of tan(2θ) using the formula: tan(2θ) = B / (A - C). In this case, tan(2θ) = 6 / (1 - 1) = 6 / 0, which is undefined.
3. Since tan(2θ) is undefined, it means that either A = C or B = 0. However, in this case, A ≠ C and B ≠ 0. Therefore, the equation cannot be solved using the tan(2θ) formula.
4. To eliminate the xy term, you can proceed with an alternative method called diagonalization. This involves finding the eigenvectors and eigenvalues of the conic matrix.
5. Write the matrix form of the conic equation: M = [A B/2; B/2 C], where A = 1, B = 6, and C = 1. The matrix is M = [1 3; 3 1].
6. Find the eigenvalues of the matrix M. To do this, find the determinant of the matrix [A - λ I], where I is the identity matrix and λ is the eigenvalue. In this case: det([1 - λ 3; 3 1 - λ]) = 0.
7. Solve the equation det([1 - λ 3; 3 1 - λ]) = 0 for the eigenvalues λ. In this case, solving the determinant equation will give you two eigenvalues: λ1 = 4 and λ2 = -2.
8. For each eigenvalue, find the corresponding eigenvector by solving the equation [A - λ I]v = 0, where v is the eigenvector. In this case:
For λ1 = 4:
[1 - 4 3; 3 1 - 4] v1 = 0
[-3 3 3; 3 -3 3] v1 = 0
Solving this system of equations will give you one eigenvector v1 = [1, -1]T.
For λ2 = -2:
[1 + 2 3; 3 1 + 2] v2 = 0
[3 3 3; 3 3 3] v2 = 0
Solving this system of equations will give you one eigenvector v2 = [1, 1]T.
9. Normalize the eigenvectors by dividing them by their magnitude. In this case, both eigenvectors already have a magnitude of 1, so no further normalization is necessary.
10. The normalized eigenvectors represent the directions of the new coordinate axes after the rotation. Therefore, the eigenvectors [1, -1]T and [1, 1]T represent the directions of the rotated axes.
11. Finally, the angle of rotation θ can be calculated by taking the inverse tangent of the slope of the eigenvector corresponding to the positive eigenvalue. In this case, θ = tan^(-1)(-1/1) = -45 degrees.
Therefore, the number of degrees the axis must be rotated to eliminate the xy term of the conic x^2+6xy+y^2-6=0 is -45 degrees.