If the axes are rotated through an angle (π)/(4) in the counter clockwise sense without shifting

If the axes are rotated through an angle (π)/(4) in the...

Key Concepts

Rotation of Axes

This concept involves changing the coordinate system by rotating it through a given angle, usually to simplify the equation of a curve or conic section. By applying rotation formulas, the new coordinates are expressed in terms of the old ones, potentially eliminating mixed terms like xy.

Elimination of the xy-Term

Conic sections in general quadratic form often contain an xy term, which can complicate their classification and analysis. By choosing an appropriate rotation angle, particularly one derived from the tangent of the mixed coefficient, the xy term can be removed, converting the equation into a simpler standard form.

Transformation Equations

These formulas relate the original coordinates to the new rotated coordinates through trigonometric functions of the rotation angle. For example, the relations x = Xcos? - Ysin? and y = Xsin? + Ycos? are used to transform the coordinates and rewrite the equation of the curve.

Conic Sections and Quadratic Forms

The study of quadratic forms in two variables encompasses the equations of conic sections such as circles, ellipses, parabolas, and hyperbolas. Understanding the structure of these equations, including coefficients of x², xy, and y², enables one to analyze and classify the geometric properties of the conic irrespective of orientation.

Transcript

00:01 We're given an equation and we are asked to find the rotation of the axes so that the new equation will contain no x y term.

00:09 And then to analyze and graph this new equation, the equation is x squared plus 4xy plus 4 y plus 5 root 5 y plus 5 equals 0.

00:27 From the previous problem, we have that the correct angle of rotation is theta equals 90.

00:35 Degrees minus one half inverse cotangent of three -fourths.

00:41 This is approximately equal to 63 .435 degrees.

00:50 And we have that the rotation formulas are x equals 1 over root 5 x prime minus 2y prime, y equals 1 of a root 5, 2x prime plus y prime.

01:11 So substitute in x and y in terms of x prime and y prime and we get that one -fifth times x -prime squared minus four x -prime y -prime plus four y -prime squared plus four x y -y this is going to be four times one -fifth or four -fifths and foiling we get times two x -prime squared plus x prime y prime minus four x prime y prime so minus three x prime y prime and minus two y prime squared plus five root or plus four y squared so this is plus four fifths in foiling we get four x prime squared plus four x prime y plus y prime squared plus 5 root 5 y this is 5 root 5 divided by root 5 is plus 5 and then 2x prime plus y prime and then moving the constant equals negative 5 and so now multiplying both sides by 5 and adding like terms we get x prime squared so 1x prime squared so 1x prime squared plus 4 times 2 is 8 x -prime squared plus 4 times 4 is 16 x -prime squared.

03:39 So we have 25x -prime squared.

03:49 The mixed term does cancel out.

03:55 For y -prime -squared, we have 4 y -prime squared minus 8 y -prime squared plus 8 y -prime squared, plus 4 y prime squared, so 0 y prime squared.

04:22 For x prime, we have 5 times 5 is 25 times 2 is 50 x prime, and for y prime, we have 5 times 5 is 25, so plus 25 y prime.

04:48 And instead of moving the constant to the other side let's keep it as plus 25 on the left side equals 0 and so one can tell immediately by looking at this because a is equal to 25 and the c is equal to 0 we have that a c equals 0 and b equals 0 as well so this is going to be a parabola however to analyze this parabola we need to use completing the square.

05:34 So first, divide through by 25 to get x prime squared plus 2 x prime plus y prime plus 1 equals 0.

05:47 And then using completing the square, we have x prime plus 1 squared.

06:00 And then this is going to be equal to negative y prime.

06:07 So that y prime is equal to equal to negative x prime plus 1 squared which we can also write as negative this is going to be four times negative 1 fourth x prime plus 1 squared better way to write this is with the square term on the left side so we have x prime plus 1 squared is equal to negative y which is also equal to negative 4 times 1 fourth i prime and so we obtain this new equation and from this we see that a is equal to one -fourth and analyzing we have that the vertex of this parabola is at zero -zero we have that the focus is going to be at zero negative a so this is going to be 0 -negative 1 fourth.

08:09 We have the directrix is going to be y prime equals a.

08:24 So y prime equals 1 fourth and we have the axis of symmetry is going to be the y prime axis and the orientation of this parabola is such that it opens down and this is going to be all in the x prime y prime plane.

09:31 This concludes the announcement of the equation.

09:35 Using this analysis, we can graph the equation.

09:42 So i start by drawing the x and y axes in black, and then i'll draw the new axes, x prime, y, and red.

09:56 I'll call the theta is about 63 .435, or about 63 .44.

10:03 So this is going to be greater than 45, or greater than 60, but less than 90.

10:10 So pretty close to 60.

10:11 Here is a good approximation.

10:45 In our analysis, we have the vertex is at 0 -0.

10:48 This lies in the parabolas, so i'll draw it in blue.

10:53 You have the focus is at 0 -9 -4 in the x -prime, y -prime plane.

10:59 So this is not on the parabolas, i'll draw it in green.

11:07 So 0, and then negative 1 4th, 0, and negative 1.