Problem 33 - Contemporary Precalculus

Question

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. circle with center (7,-4) and radius 6

Step-by-Step Solution

Answer

Question: Write the parametric equations of a circle with a center at (7, -4) and a radius of 6, and confirm your answer by graphing the circle. Answer: The parametric equations of the circle are x(t) = 7 + 6*cos(t) and y(t) = -4 + 6*sin(t), with t ranging from 0 to 2*pi. Graphing these equations will confirm the correctness of our answer by showing a circle centered at (7, -4) with a radius of 6.

Step 1: Write the equation of the circle

Using the general formula of a circle: (x - h)² + (y - k)² = r², where (h, k) is the center, and r is the radius. For our circle, the center is (7, -4) and the radius is 6. So, the equation will be (x - 7)² + (y + 4)² = 6².

Step 2: Parameterize the equation of the circle

We parameterize the circle by introducing an angle parameter t, with the coordinates x and y expressed in terms of cosine and sine, respectively. x = h + r*cos(t) y = k + r*sin(t) Plugging in the center coordinates (7, -4) and the radius 6 into the parameterized equations, we get: x = 7 + 6*cos(t) y = -4 + 6*sin(t) So, the parametric equations of the circle are: x(t) = 7 + 6*cos(t) y(t) = -4 + 6*sin(t), with t ranging from 0 to 2*pi.

Step 3: Confirm the answer by graphing the circle

To confirm the correctness of the parametric equations, plot the circle using parametric equations x(t) = 7 + 6*cos(t) and y(t) = -4 + 6*sin(t) for t ranging from 0 to 2*pi. The circle with center (7, -4) and radius 6 should be graphed correctly, confirming the correctness of our answer.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. Depending on the angle of the intersection, we can get different shapes such as circles, ellipses, parabolas, and hyperbolas. These curves have been extensively used in mathematics, physics, and engineering to model various phenomena.

In the context of the provided exercise, we are dealing with a circle, one of the simplest conic sections. A circle is a special case of an ellipse where the two foci coincide, resulting in a symmetrical shape. Circles can be described using a standard equation, and in our example, the circle has clearly defined parameters like its center and radius. Understanding conic sections helps in visualizing and solving equations involving these kinds of shapes.

Studying conic sections requires familiarity with their respective properties, which enable mathematicians to predict and portray these shapes accurately. This understanding is crucial for solving exercises like the one we encountered in this task.

Circle Equation

The circle equation is fundamental in identifying and working with circles in geometry. These equations help in defining the entire set of points that make up the shape.

The standard equation of a circle is \[(x - h)^2 + (y - k)^2 = r^2\] where \((h, k)\) represents the center, and \(r\) the radius of the circle. The equation states that every point on the circle's periphery is a fixed distance \(r\) from the center.

For our specific problem, the center \((h, k)\) is given as \((7, -4)\) and the radius \(r\) is 6. Plugging these into the formula, the circle equation becomes \[(x - 7)^2 + (y + 4)^2 = 36\] This equation serves as the foundation for further operations like parameterization.

Identifying and using the circle equation is essential for graphing and solving geometrical problems accurately.

Trigonometry

Trigonometry plays a pivotal role in the parameterization of circles. By using the trigonometric functions cosine and sine, we can express the circle's equation in terms of a parameter \(t\). This approach is particularly useful in translating geometric shapes into an analytical form.

The coordinates are parameterized as:

• \(x = h + r \cos(t)\)
• \(y = k + r \sin(t)\)

Here, \(t\) represents the angle made with the positive \(x\)-axis, ranging from 0 to \(2\pi\) for a full circle.

In our example, substituting \((h, k) = (7, -4)\) and \(r = 6\) gives the parametric equations:

• \(x(t) = 7 + 6\cos(t)\)
• \(y(t) = -4 + 6\sin(t)\)

Through trigonometry, we effectively transform a static circle equation into a dynamic representation that can be used in graphing and simulations.

Understanding trigonometry is essential for visualizing the movement along the circle, providing insights into complex calculations and graphs.

Graphing

Graphing is the ultimate step in visualizing and verifying the correctness of mathematical equations. After parameterizing our circle, plotting it helps confirm the solution's accuracy. By inputting the parametric equations into graphing software or plotting manually, you can see how the circle takes shape on a coordinate plane.

For the parametric equations \(x(t) = 7 + 6\cos(t)\) and \(y(t) = -4 + 6\sin(t)\), and by letting \(t\) range from 0 to \(2\pi\), a full circle should appear centered at \((7, -4)\) with a radius of 6.

Graphing not only helps in checking answers but also enhances spatial understanding of how mathematical equations translate into visual representations. It’s a powerful tool in learning and retaining knowledge, especially in geometry.