Transform the given equations by rotating the axes through the given angle. Identify and sketch

step 1 Hello, welcome to this lesson. In this lesson, we transformed this equation. Then after that, uh

Transform the given equations by rotating the axes through...

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Rotation of Axes

This concept involves performing a coordinate transformation to simplify the equation of a conic by rotating the coordinate system. By using transformation formulas such as x = x' cos? - y' sin? and y = x' sin? + y' cos?, the new coordinates (x', y') can be used to realign the axes so that the conic takes on a simpler form, typically without any xy (cross-product) term.

Elimination of the xy-term

In the general quadratic equation of a conic, the xy-term indicates that the conic is rotated relative to the coordinate axes. By choosing an appropriate rotation angle, determined by the relation tan(2?) = B/(A - C) where A, B, and C are the coefficients from the quadratic form, one can eliminate this cross term, thus simplifying the equation and making it easier to identify and graph the conic section.

Conic Sections

Conic sections are curves that can be defined as the intersection of a plane with a double-napped cone and are represented by quadratic equations in two variables. The classification (ellipse, hyperbola, or parabola) depends on the discriminant derived from the coefficients of the quadratic form. Transforming the conic into its canonical form, by eliminating the xy-term and possibly completing the square, reveals its inherent geometric nature and properties.

Transcript

00:01 Hello, welcome to this lesson.

00:01 In this lesson, we transformed this equation.

00:06 Then after that, uh, transform it by retreating through this angle.

00:13 Okay.

00:14 Then after that we identify the kind of equation that we have.

00:21 So we have this and let's identify than transform later.

00:27 So we compare to the general form.

00:44 Then we pick a determinant that is 4a minus b minus 4a c.

00:53 So if this is equal to 0, then we have a parabola.

01:01 If this is greater than 0, we have a hyperbola.

01:10 And if this is less than 0, we have an ellipse.

01:15 So looking at it, we have the b, which is a coefficient of x and y as negative 4.

01:28 We have a, which is a coefficient of x squared as 8.

01:33 Then we have the c, which is a coefficient of y squared as 5.

01:39 And this gives us, this is 16 minus 160, so 1, negative 144, which is let's let's.

01:50 And zero so we have an ellipse okay that is the nature okay now let's go to the transformation so we have y that is equal to x that is equal to x prime cost thicker minus y prime sine theta so this implies that x is equal to now the cost theta of the angle given is square root of 5 on 5.

02:35 So we have square root of 5 on 5 x prime.

02:41 Then the sine theta of the angle, we have 2 on 5, square root of 5.

02:46 So that is 2 square root of 5, 5...

Please give Ace some feedback