(a) Find parametric equations for the ellipse x^2 / a^2+y^2...

(a) Find parametric equations for the ellipse $x^2 / a^2+y^2 / b^2=1$. [Hint: Modify the equations of the circle in Example 2.]
(b) Use these parametric equations to graph the ellipse when $a=3$ and $b=1,2,4$, and 8 .
(c) How does the shape of the ellipse change as $b$ varies?

Key Concepts

Parameter Variation

Adjusting the parameters in the ellipse's equation, particularly the semi-minor axis parameter b while keeping the semi-major axis a constant, affects the shape of the ellipse. As b increases or decreases, the ellipse becomes more or less stretched in the y-direction, respectively, altering the relative proportions of the ellipse without changing its overall center or orientation.

Graphing Parametric Curves

Graphing parametric curves involves plotting the set of points defined by the parametric equations over a range of the parameter. This method is particularly useful for ellipses, where varying the parameter t from 0 to 2? produces the complete curve, and it can visually illustrate how changes to the parameters a and b affect the shape.

Ellipse

An ellipse is a conic section defined by a quadratic equation where the sum of distances from any point on the ellipse to two fixed points (foci) is constant. Its standard form equation is given by x²/a² + y²/b² = 1, which represents an ellipse centered at the origin with semi-major and semi-minor axes depending on the values of a and b.

Parametric Equations

Parametric equations express the coordinates of the points on a curve as functions of a separate parameter, typically denoted by t. This approach allows one to describe curves in the plane, such as circles and ellipses, in a way that emphasizes how the points change with the parameter, enabling easy graphing and analysis of the shape.

Transformation from Circle to Ellipse

A common method to derive the parametric equations for an ellipse involves starting with the parametric representation of a circle, x = cos(t) and y = sin(t), and then applying a scaling transformation. Multiplying the x-coordinate by a and the y-coordinate by b transforms the circle into an ellipse, yielding x = a cos(t) and y = b sin(t).

Transcript

00:01 We want to find a parameterization for the ellipse, x squared over a squared plus y squared over b squared equals one.

00:09 So for a circle, we usually use the parameterization x and cosine theta, y equals sine theta, and then we get the identity cosine squared theta plus sine squared theta equals one.

00:20 We can't do that here, though, because we're left with this a squared term and this b squared term, which means we can't use that identity.

00:29 So we want to introduce an a term to x, which will cancel out this a squared, and a b term to y, which will cancel out this b squared.

00:40 So let's try.

00:41 X is equal to a, a to cancel out the a in the denominator, cosine t to be used with the identity.

00:50 Y is equal to b, sine t.

00:53 Then we have x squared over a squared plus y squared over b squared over b squared.

01:00 Is equal to a cosine t squared over a squared plus b sine t squared over b squared.

01:15 The a squared will cancel out as will the b squared, which leaves us with cosine squared t plus sine squared t, which is equal to one.

01:27 So this does parameterize the ellipse.

01:32 Now we're going to look at a few examples of this.

01:36 Oh, very important to mention is the interval of parameterization.

01:40 Similarly to a circle, the period of cosine and sine is 2 pi...

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