x^2+16xy+y^2-8=0 Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term
To eliminate the x'y'-term in the equation x^2 + 16xy + y^2 - 8 = 0, we can apply a rotation transformation to the coordinate system. Here are the steps to do that:
Step 1: Calculate the angle of rotation (θ) needed to eliminate the x'y'-term by using the formula:
tan(2θ) = 2Cov(x, y) / (Var(x) - Var(y))
Where Cov(x, y) represents the covariance between x and y, and Var(x) and Var(y) represent the variances of x and y respectively.
Step 2: Substitute the angle of rotation (θ) into the following rotation formulas:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Step 3: Rewrite the equation using the new variables (x' and y') to get the rotated equation.
By applying these steps, you should be able to rotate the coordinate system and eliminate the x'y'-term in the equation.
To remove the x'y'-term in the equation x^2 + 16xy + y^2 - 8 = 0, we need to perform a rotation of axes.
We can follow the following steps to obtain the rotation formulas:
Step 1: Calculate the angle of rotation (θ) using the formula:
θ = 0.5 * atan(2 * B / (A - C))
where A, B, and C are coefficients of x^2, xy, and y^2 respectively. In this case, A = 1, B = 16, and C = 1.
Step 2: Substitute the known values into the formula to find θ:
θ = 0.5 * atan(2 * 16 / (1 - 1))
θ = atan(32)
Step 3: Determine the new coordinates x' and y' in terms of the original coordinates x and y using the rotation formulas:
x' = x * cos(θ) + y * sin(θ)
y' = -x * sin(θ) + y * cos(θ)
Finally, applying the rotation transformation to the given equation x^2 + 16xy + y^2 - 8 = 0, we get:
(x * cos(θ) + y * sin(θ))^2 + 16(x * cos(θ) + y * sin(θ))( -x * sin(θ) + y * cos(θ)) + (-x * sin(θ) + y * cos(θ))^2 - 8 = 0
Simplifying this equation will eliminate the x'y'-term and result in a rotated system.