00:02
You're given an equation and we're asked to rotate the axis to eliminate the xy term in this equation.
00:09
Then we're asked to write the equation in the standard form and then to sketch the graph of the resulting equation showing both sets of axes.
00:17
The equation is 2x squared plus xy plus 2y squared minus 8 equals 0.
00:30
So in order to rotate the axes, first we'll determine the angle by which to rotate.
00:40
From our formulas from this section, you know that the angle should satisfy the cotangent of 2 theta is equal to a, the coefficient of x squared, which is 2, minus the coefficient of y squared, which is 2, over the coefficient of x, y, which is 1.
00:58
This is 0 over 1 or 0, so that the 2 theta should be equal to 2.
01:04
To pi over 2, and therefore theta should be equal to pi over 4 or 90 degrees, sorry 45 degrees and this is counterclockwise.
01:16
And then to obtain the new coefficients, we'll make the substitutions x equals x prime cosine of theta or pi over 4 minus y prime sine of theta or pi over 4.
01:29
This simplifies to root 2 over 2 times x prime minus y prime.
01:38
And y equals x prime sine of theta or pi over four plus y prime cosine of theta or pi over four.
01:49
This is equal to root 2 over 2 times x prime plus y prime.
01:58
Substituting we get 2 times root 2 over 2 squared, which is the same as 1 1ā2 times x prime minus y prime squared, plus root 2 over 2 times x prime minus y prime times root 2 over 2 times x prime plus y prime plus y prime plus y prime plus 2 plus 2 plus y prime squared and then simplifying this is x prime plus y prime squared prime y prime plus y prime squared plus root 2 over 2 times root 2 over 2 is one half x prime squared minus y prime squared plus x prime squared plus 2x prime y prime plus y prime squared minus 8 equals 0 so now i'm going to do two so first i'll simplify.
03:44
We have 2x prime squared minus 1 1 half x prime squared.
03:48
Let's actually multiply by 2 first.
03:51
So we have 2x prime squared plus x prime squared plus 2x prime squared is 5 times x prime squared.
04:01
And then we have negative 4x prime y prime plus 4x prime y prime is 0x prime y prime prime.
04:08
We also have plus 2 y prime squared minus y prime squared plus 2 y prime squared plus 2 y prime is plus 3 y prime squared, and then we have left over minus 16 equals 0.
04:23
At this point, we have eliminated the mixed term x prime y prime, but we still want to write this in standard form.
04:30
So i'll isolate by moving 16 to the other side and divide by 16, and we get x prime squared over 165ths plus y prime squared over 16 thirds equals 1.
04:53
And so this is the equation in standard form.
04:58
From this form, it's easy to see that this is the equation of an ellipse with the center at the origin.
05:07
This is an x prime, y, prime, coordinates, by the way...