Problem 15 - Calculus

Question

Convert \(r=6 \sin \theta+8 \cos \theta\) to the \(x y\) equation of a circle (what radius, what center?).

Step-by-Step Solution

Answer

Center: (4, 3), Radius: 5.

Step 1: Use Polar to Cartesian Conversion

First, recall the conversion formulas from polar to Cartesian coordinates: \ \(x = r \cos \theta\) \ \(y = r \sin \theta\). To transform our given equation \(r = 6 \sin \theta + 8 \cos \theta\) into Cartesian form, express \(\sin \theta\) and \(\cos \theta\) in terms of \(x\) and \(y\). Use \(\sin \theta = \frac{y}{r}\) and \(\cos \theta = \frac{x}{r}\). Substituting these results in, we get: \ \(r = 6 \frac{y}{r} + 8 \frac{x}{r}\).

Step 2: Simplify to Isolate r

Multiply through by \(r\) to eliminate the fractions, giving: \ \(r^2 = 6y + 8x\). Express \(r^2\) in terms of \(x\) and \(y\) using \(r^2 = x^2 + y^2\). This will yield the equation: \ \(x^2 + y^2 = 6y + 8x\).

Step 3: Rearrange to Complete the Square

Rearrange the equation to group the \(x\) terms and \(y\) terms: \ \(x^2 - 8x + y^2 - 6y = 0\). Complete the square for both \(x\) and \(y\): \ \, \((x - 4)^2 = x^2 - 8x + 16\) \ \((y - 3)^2 = y^2 - 6y + 9\). Add these squares to balance the equation: \ \((x - 4)^2 + (y - 3)^2 = 16 + 9 = 25\).

Step 4: Identify the Circle Information

Now the equation is in the standard form of a circle \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Here, \((h, k) = (4, 3)\) and \(r^2 = 25\), hence \(r = 5\). The circle has a center at \((4, 3)\) and a radius of 5.

Key Concepts

Equation of a Circle

An equation of a circle in Cartesian coordinates presents a clear and structured way to represent a circle on a plane. The general form of this equation is \( (x - h)^2 + (y - k)^2 = r^2 \)
where:
\( (h, k) \) is the center of the circle.
\( r \) is the radius of the circle.
This formula helps in visualizing the circle’s position and how far points on the circumference are from its center. The importance of using this form is its simplicity and how it succinctly provides crucial information about the circle.

For instance, if you have an equation such as \( (x - 4)^2 + (y - 3)^2 = 25 \), this tells you that the center of the circle is at \((4, 3)\) and the radius is \( 5 \) since \( r^2 = 25 \). By simply knowing these parameters, anyone can easily sketch the circle or understand its orientation in the Cartesian plane.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into a form that easily reveals certain key properties. This approach is especially handy with circles.

For the expression \( x^2 - 8x \), completing the square involves reshaping it into \( (x - 4)^2 \). Here are the steps:

• Take half of the coefficient of \( x \). This is \( -8/2 = -4 \).
• Square the result: \( (-4)^2 = 16 \).
• Add and subtract 16 in the equation: \( x^2 - 8x + 16 - 16 \).

Through this process, the equation transforms into \( (x - 4)^2 - 16 \).

Repeating this process for the \( y \) terms allows you to similarly complete the square for \( y^2 - 6y \). The transformation helps to rearrange the circle's equation into its standard form, making it easy to identify the center and radius.

Cartesian Coordinates

Cartesian coordinates are fundamental in depicting points in a two-dimensional plane using a pair of values \( (x, y) \). Each point on the plane is mapped uniquely with these two numbers, symbolizing its position relative to the origin \((0, 0)\).

An advantage of this system is its ability to precisely define geometric shapes, like lines and circles, via equations.
For a circle, using Cartesian coordinates makes it easy to interpret spatial relationships and distances.
A point \((x, y)\) lies on a circle if its coordinates satisfy the circle's equation. This concreteness aids in various graphical and analytical tasks, such as plotting or integrating other mathematics concepts. Whether you are working on geometry or solving real-world problems, Cartesian coordinates serve as a versatile tool in navigation and analysis of the plane.