A curve, called a witch of Maria Agnesi, consists of all...

A curve, called a witch of Maria Agnesi, consists of all possible positions of the point $P$ in the figure. Show that parametric equations for this curve can be written as

$$
x=2 a \cot \theta \quad y=2 a \sin ^2 \theta
$$

Sketch the curve.
(GRAPH CAN'T COPY)

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Key Concepts

Parametric Equations

Parametric equations describe a curve by expressing the coordinates (x and y) as functions of a parameter (often denoted by ? or t). This approach can provide insights into the curve's behavior and properties that might be less obvious when using a single Cartesian equation.

Trigonometric Functions

Trigonometric functions are vital in linking angles to coordinates in problems involving circular or periodic phenomena. In the context of parametric curves, functions such as sine, cosine, and cotangent allow for the description and manipulation of curves defined by angular parameters.

Curve Sketching

Curve sketching is the process of drawing a curve by identifying key features such as intercepts, asymptotes, and regions of increase or decrease. For curves defined parametrically, this involves analyzing how changes in the parameter affect the shape and position of the curve.

Witch of Maria Agnesi

The Witch of Maria Agnesi is a historically significant mathematical curve with distinct geometric properties. Its construction involves special parametric equations derived using trigonometric functions, exemplifying the interplay between algebraic methods and geometric intuition in understanding complex curves.

Transcript

00:01 We're going to find parametric equations that represent the curve generated by points such as p.

00:07 So we're first going to find the x coordinate.

00:12 So notice that the height of this point is the same as the height of the circle, which is 2a.

00:19 And let's drop this down a little bit.

00:24 2a.

00:26 Well, considering this right triangle here, we have the length of one side and this angle thing.

00:32 So then we want to find this side here, which is the same as the x coordinate of p.

00:40 So we have x is equal to 2a.

00:44 Well, we want adjacent over opposite or cotangent of theta.

00:51 That's the first parameter.

00:54 Now we'll get y.

00:56 So for the y coordinate, it's the same as the height of this triangle here, right here.

01:04 But we don't know any side length of this.

01:08 However, notice that if we go over here to this triangle, we do know this angle, which is pi over 2 minus theta, and we know the length of this side, which is 2a, so we can get the length of this side of the triangle, and then use that information with this triangle here to find the length of the side we want, which will give us y.

01:39 So let's start over here.

01:42 The length of this line is equal to 2a times we want adjacent over hypotenuse, which is cosine of pi over 2 minus theta...

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