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 will transform the given equation by rotating

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

Key Concepts

Rotation of Axes

This concept involves applying a coordinate transformation to rewrite an equation in terms of new variables. The transformation uses relationships such as x = x'cos? – y'sin? and y = x'sin? + y'cos?, allowing us to rotate the coordinate system by an angle ?. This procedure is essential in reducing complicated equations into more manageable forms or aligning them with standard positions.

Invariant Properties of Circles Under Rotation

A circle is characterized by its symmetry, and its equation remains unchanged under rotation about its center. This means that when a circle’s equation, like x² + y² = 16, is transformed via a rotation of axes through any angle, its fundamental attributes—center and radius—remain constant. This invariance simplifies the analysis and sketching of circles after transformation.

Transcript

00:01 Hello, welcome to this lesson.

00:01 In this lesson, we will transform the given equation by rotating it through an angle of 60 degrees, then we would identify the kind of equation that is formed.

00:14 So with this one, we have x, that is equal to x prime, cost theta minus y prime, sine theta, then y that is equal to xx, then y, that is equal to x prime, sine theta, plus y, and then y, and then y that is equal to x prime, and then y, and then.

00:36 Prime cost theta.

00:43 Ok, so this one would placing the values of the cost theta.

00:52 So that is equal to y prime.

00:56 Then cost 60 is half.

01:00 So we have this on 2 minus y prime.

01:05 Sin theta is square width of 3 on 2.

01:10 So we have square root of 3 on 2.

01:14 Y prime then this one we have y that is equal to square root of 3 on 2 x prime plus y prime prime on 2 okay so x squared plus y squared is called to 16 we will put in the values so you have x 1 on 2 minus square root of 30 on 2 y prime squared then we have square root of 3 on 2 x prime plus y prime on 2 squared squared squared so with this one we have x squared on 2 r squared then minus 2 times x squared of 3 on 4 times again.

02:58 Okay, let me bring these ones here.

03:18 Okay, then we have a positive root of 3 on 4.

03:29 Or this is what i will make it squared.

03:37 That is the expansion of the first one...

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