Question

13 x²+6 √3 x y+7 y²=16 Use a rotation of axes to eliminate the x y -term.

Answer 1

To eliminate the xy-term in the equation 13x² + 6√3xy + 7y² = 16 using a rotation of axes, we need to find an angle of rotation that makes the xy-term disappear. Once we have the angle of rotation, we can use the rotation matrix to find the new coordinates (x', y') in terms of the original coordinates (x, y). Using the rotation matrix, we find that x' = xcosθ - ysinθ and y' = xsinθ + ycosθ. Substituting these values into the original equation, we get the new equation in terms of x' and y' where the xy-term is eliminated.

Step-by-step explanation:

To eliminate the xy-term in the equation 13x² + 6√3xy + 7y² = 16 using a rotation of axes, we need to find an angle of rotation that makes the xy-term disappear. This can be done by setting the angle of rotation as the solution to the equation tan(2θ) = √3/13.

Once we have the angle of rotation, we can use the rotation matrix to find the new coordinates (x', y') in terms of the original coordinates (x, y).

Using the rotation matrix, we find that x' = xcosθ - ysinθ and y' = xsinθ + ycosθ. Substituting these values into the original equation, we get the new equation in terms of x' and y' where the xy-term is eliminated.