rotate the axes and eliminate the xy term of 3(x^2)+2xy+3(y^2)-8
To rotate the axes and eliminate the xy term of the given equation, we can follow these steps:
1. Determine the angle of rotation: The angle of rotation needed to eliminate the xy term can be found using the formula: theta = 0.5 * arctan(2 * coefficient of xy / (coefficient of x^2 - coefficient of y^2)). In this case, the coefficient of xy is 2, the coefficient of x^2 is 3, and the coefficient of y^2 is also 3. Plugging these values into the formula gives us: theta = 0.5 * arctan(2 * 2 / (3 - 3)) = 0.5 * arctan(4 / 0) = 90°.
2. Apply the rotation matrix: Rotate the xy-coordinate system counterclockwise by the angle of rotation theta. The rotation matrix for a counterclockwise rotation of theta is:
| cos(theta) -sin(theta) |
| sin(theta) cos(theta) |
Since theta = 90°, the rotation matrix becomes:
| 0 -1 |
| 1 0 |
3. Substitute the new coordinates: To eliminate the xy term, substitute new variables u and v, where u = x * cos(theta) - y * sin(theta) and v = x * sin(theta) + y * cos(theta). Substitute these values into the original equation.
The original equation: 3(x^2) + 2xy + 3(y^2) - 8
Substituting u and v into the equation gives us:
3[(u * cos(theta) - v * sin(theta))^2] + 2(u * cos(theta) - v * sin(theta))(u * sin(theta) + v * cos(theta)) + 3[(u * sin(theta) + v * cos(theta))^2] - 8
Simplify this expression.
4. Identify u-v equation terms: Look for terms in the expression resulting from the substitution of u and v that are of the form a(u^2) + b(uv) + c(v^2). These terms will be linear combinations of u^2, uv, and v^2.
In our equation, the terms will have coefficients of sin(theta) and cos(theta). Collect all the terms that have either sin(theta) or cos(theta) as their coefficient.
5. Eliminate the uv term: If there is an uv term present after collecting the terms, it means the rotation was not completely successful in eliminating the xy term. In this case, reassign new variables x' and y' to eliminate the uv term. The new variables x' and y' can be found by using the equations x = x' * cos(theta) + y' * sin(theta) and y = -x' * sin(theta) + y' * cos(theta).
6. Final form: After eliminating the uv term, we should be left with an equation containing only terms of the form d(x'^2) + e(y'^2). The equation will represent a conic section with its axes aligned with the new u-v or x'-y' coordinate system.
Following these steps should help you to rotate the axes and eliminate the xy term of the given equation.