ROtate the conic so that its equation 3xy-4y²+ 18=0 has no xy term

To rotate a conic so that it has no xy term, you can use the concept of orthogonal coordinate transformation. Here's how you can do it step by step:

Step 1: Identify the coefficients of the xy term, y² term, and constant term in the given equation. In this case, the equation is 3xy - 4y² + 18 = 0, and the coefficients are:
- The coefficient of the xy term is 3.
- The coefficient of the y² term is -4.
- The constant term is 18.

Step 2: Calculate the angle of rotation (θ) using the formula:
θ = (1/2) * arctan((2 * coefficient of xy term) / (coefficient of y² term - coefficient of x² term))

In this case, the coefficient of x² term is 0 because there is no x² term in the given equation. Therefore, the angle of rotation can be calculated as follows:
θ = (1/2) * arctan((2 * 3) / (-4 - 0))
θ = (1/2) * arctan(-6 / -4)
θ = (1/2) * arctan(3/2)

Step 3: Once you calculate the angle of rotation (θ), you can create a new coordinate system by rotating the original coordinates (x, y) using the formula:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)

In this case, the original equation is 3xy - 4y² + 18 = 0, and the new coordinates (x', y') can be substituted into the equation:
3(x' * cos(θ) - y' * sin(θ))(x' * sin(θ) + y' * cos(θ)) - 4(y' * cos(θ) + x' * sin(θ))(y' * cos(θ) + x' * sin(θ)) + 18 = 0

Step 4: Simplify and rearrange the equation to eliminate the xy term.

By multiplying and simplifying the equation from Step 3, you will eventually obtain a new equation without the xy term.

And that's how you can rotate the conic so that its equation has no xy term.