Problem 35 - Thomas Calculus

Question

The witch of Maria Agnesi The bell-shaped witch of Maria Agnesi can be constructed in the following way. Start with a circle of radius \(1,\) centered at the point \((0,1),\) as shown in the accompanying figure. Choose a point \(A\) on the line \(y=2\) and connect it to the origin with a line segment. Call the point where the segment crosses the circle \(B\) . Let \(P\) be the point where the vertical line through \(A\) crosses the horizontal line through \(B\) . The witch is the curve traced by \(P\) as \(A\) moves along the line \(y=2\) . Find parametric equations and a parameter interval for the witch by expressing the coordinates of \(P\) in terms of \(t\) the radian measure of the angle that segment \(O A\) makes with the positive \(x\) -axis. The following equalities (which you may assume) will help. $$\begin{array}{ll}{\text { a. } x=A Q} & {\text { b. } y=2-A B \sin t} \\\ {\text { c. } A B \cdot O A=(A Q)^{2}}\end{array}$$

Step-by-Step Solution

Answer

The parametric equations are \( x(t) = \frac{2\cos t}{1+\sin t} \) and \( y(t) = 1 + \sin t \) with \( t \) in \((-\pi/2, \pi/2)\).

Step 1: Understand the Problem Setup

To solve this problem, we need to determine the parametric equations for the curve traced by point \( P \) as point \( A \) moves along the line \( y = 2 \). The circle is centered at \((0,1)\) with radius 1. Point \( A\) is on the line \( y = 2 \), and point \( B \) is where the line from \( O \) (the origin) through \( A \) intersects the circle. Point \( P \) is found by intersecting the vertical line through \( A \) with the horizontal line through \( B \). The goal is to express \( P \) in parametric form using the angle \( t \).

Step 2: Set Up the Geometry

Choose point \( A \) on the line \( y=2 \). Connect point \( A(x,2) \) to the origin \((0,0)\). Call the point \( B(x_B, y_B) \) where the line \( OA \) intersects the circle. Since the circle's equation is \((x_B-0)^2 + (y_B-1)^2 = 1^2\), the segment \( OA \) has the equation \( y = (2/x)x \).

Step 3: Determine the Intersection at B

Using the parametric angle \( t \), let the coordinates of point \( B \) be on the circle such that the line defined by segment \(OA\) intersects the circle. Thus, at point \( B \), \(x_B = \cos t\) and \(y_B = 1 + \sin t\), satisfying the circle equation \((cos t)^2 + (sin t)^2 = 1\).

Step 4: Derive the Coordinates of A

Let's express \( A \) using the given equation: let \( A = (x, 2) \), with \( x = AQ = AB \cos t\). From equality \(c\), \(OA = \sqrt{x^2 + 4}\) and thus \( AB \cdot OA = x^2\), giving us \( AB = \dfrac{x^2}{\sqrt{x^2+4}} \). Solve for \( x \) gives \( x = \dfrac{2\cos t}{1+\sin t} \).

Step 5: Find Parametric Equation for P

The horizontal line through \( B \) is \( y = 1 + \sin t \) and the vertical line through \( A \) is \( x = \dfrac{2\cos t}{1+\sin t} \). The coordinates of \( P \) is where these lines intersect, \( (\dfrac{2\cos t}{1+\sin t}, 1 + \sin t) \).

Step 6: Determine Parameter Interval

For \( A \) to trace along the line \( y = 2 \), \( t \) should vary from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), as this covers points where the angle between \( OA \) and the positive x-axis allows the line to travel through all possible positions along \( A \) on \( y = 2 \).

Key Concepts

Geometry

Geometry provides a way to understand shapes, sizes, and the relative position of figures in space. In our context, it helps in examining the movement and transformation of geometric entities, such as points, lines, and circles. For this exercise, we begin with a fundamental geometric shape: the circle. This circle, centered at \((0, 1)\) with a radius of \(1\), creates a crucial frame of reference.

• Every point on the circle satisfies the circle equation: \((x - 0)^2 + (y - 1)^2 = 1^2\).
• Line segments and intersection points form the building blocks for understanding parametric equations.
• The circular path guides the path trace defined by the point \(P\), through which we find the path of movement as it varies on the line \(y = 2\).

When you think of geometry here, imagine how we map a path of movement and connect points to curves in space, creating a more visual representation of the problem.

Circle Equations

Circle equations are pivotal in determining the relationship between different points across a circular path. In essence, any point on this circle must adhere to a set mathematical equation characteristic of circles: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center, and \(r\) is the radius.

For this problem:

• Our circle's center is at \((0, 1)\), and its radius is \(1\).
• The equation becomes \(x^2 + (y - 1)^2 = 1\).

This simplifies analysis, as every intersection involving the circle ensures all points \(B\) remain on it. As the segment \(OA\) intersects the circle, using the parametric angle \(t\), we derived that the coordinates for point \(B\) were \((\cos t, 1 + \sin t)\). This realization ensures that no matter where \(A\) lies on the line \(y=2\), point \(B\) will always satisfy the circle's equation.

Angle Measurement

An understanding of angles and their measurement is essential to navigate through the path traced by point \(P\).

• We consider the angle \(t\), which is the angle made by the line segment \(OA\) with the positive \(x\)-axis.
• This angle \(t\) becomes deeply intertwined with the positions of \(A, B,\) and \(P\), serving as a linchpin between trigonometric relations and the parametric equations.

By employing trigonometric identities, we represent points and equations in terms of \(\cos t\) and \(\sin t\) – vital components in the conversion from regular to parametric forms. The angle range from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) reflects all possible inclinations of line \(OA\) intersecting the circle within the specific bounds of the problem. As we shift \(t\), \(A\) correspondingly moves along the line \(y=2\), guiding the path of \(P\) and illustrating just how flexible and interconnected these geometric relationships can be.