Rotations of Conics

This is one of those days where Miss V decided to make my life very difficult. But thats every day. Also she doesn't text us when we have a trig quiz. Im grumpy. Jk I love Miss V. Anyways. This is re-writing an equation to get a shift in the x and y axis. You need about 4 equations to be successful in this and a lot of paper because these problems are long. To solve these problems follow these steps: 

1. Solve for angle using cot(theta) = a-c/b use your unit circle to find the angle degree. 

2. Find the new x and y using the two equations below. These will give you x prime and y prime 

3. Substitute new x and y into the original equations. If this works the xy should cancel out. Be very neat becuase there is a lot of work. 

So far, we have only studied conics whose axes were parallel to the coordinate axes. Such conics can be written with the equation Ax ² + Cy ² + Dx + Ey + F = 0 . The coefficient of the xy term, B , equals zero when the conic's axes are parallel to the coordinate axes. When the axes of the conic are not parallel to the coordinate axes, we can still express the conic without using an xy term by rotating the coordinate axes to from a new plane. Instead of using the xy-plane, the conic can be expressed as a second degree equation ( Ax ² + Cy ² + Dx + Ey + F = 0 ) in thex'y' -plane. The new second degree equation of the conic, after the axes have been rotated, will look like this: A'x' ² + C'y' ² + D'x' + E'y' + F' = 0 . The advantage to doing this is that by avoiding an xy -term, we can still express the equation of the conic in standard form.

When a conic contains an xyterm, the x and y axes can be rotated through an angle of θsuch that they are once again parallel with the axes of the conic, thus eliminating the xyterm of the conic. A rotation of the coordinate axes looks something like this:

Figure %: The x and y axes are rotated through an angle of θ to their new position as the x' and y' axes.The newly rotated axes are called the x' and y' axes.

A point in the x'y' -plane can be expressed with x and y coordinates, and vice-versa, using the following conversion formulas:

x = x'cos(θ) - y'sin(θ)

y = x'sin(θ) + y'cos(θ)

x' = x cos(θ) + y sin(θ)

y' = - x sin(θ) + y cos(θ)

Here's an example problem: what are the x and y coordinates of the point (x',y') = (1, 0) in the x'y' -plane which is the xy -plane rotated an angle of radians? Using the formulas above, x = 1 cos() - 0 sin() =  , and y = 1 sin() + 0 sin() =  . So the x and y coordinates of this point are (,) .

To eliminate the xy term of a conic of the form Ax ² + Bxy + Cx ² + Dx + Ey + F = 0 in order to use its standard form and write it in an equation of the form A'x' ² + C'y' ² +D'x' + E'y' + F' = 0 , you must rotate the coordinate axes through an angle θ such that cot(2θ) =  . Once this is done, and θ is found, you can use the conversion equations above to change the x and y coordinates to x' and y' coordinates. Then standard form can be used and the foci and vertices will be easily recognizable--but they will be in the x' and y' coordinate system. When they are found, it is possible to convert back to x and y coordinates to sketch the conic.