Problem 63 - Thomas Calculus

Question

Find polar equations for the circles in Exercises \(57-64 .\) Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. $$x^{2}+y^{2}+y=0$$

Step-by-Step Solution

Answer

Polar equation: \(r = \frac{1}{4} - \sin \theta\).

Step 1: Rewrite the Equation

Start with the given equation of the circle in standard Cartesian form: \(x^{2} + y^{2} + y = 0\). The goal is to express it in a form that is useful for conversion to a polar equation.

Step 2: Complete the Square for y

We complete the square for the \(y\) term in the equation. Split the \(y\) term from the rest: \(x^{2} + (y^{2} + y) = 0\). Take half of the coefficient of \(y\), square it, and add and subtract it inside the equation. This results in: \(x^{2} + (y + \frac{1}{2})^2 - \frac{1}{4} = 0\).

Step 3: Simplify the Equation

Add \(\frac{1}{4}\) to both sides to balance the equation: \(x^{2} + (y + \frac{1}{2})^2 = \frac{1}{4}\). It represents a circle with center at \((0, -\frac{1}{2})\) and radius \(\frac{1}{2}\).

Step 4: Convert to Polar Coordinates

The polar coordinates \(x = r\cos\theta\) and \(y = r\sin\theta\) can be substituted into the equation. Substitute these into the adjusted equation: \(r^{2} = \frac{1}{4} - r \sin \theta\), yielding a polar equation for this circle.

Step 5: Identify Polar Equation

The polar equation rearranges as \(r = \frac{1}{4} - \sin \theta\) because of the circle's properties. The circle is shifted down by \(-\frac{1}{2}\) in the Cartesian plane, corresponding to adjusting the polar equation.

Step 6: Sketch the Circle

Sketch both the Cartesian and polar representations. The circle is centered at \((0, -\frac{1}{2})\) with radius \(\frac{1}{2}\). The polar sketch should match the circle's shape described by \(r = \frac{1}{4} - \sin \theta\).

Step 7: Label the Circle

Label the sketched circle with its equations. The Cartesian equation is \(x^{2} + y^{2} + y = 0\), and the polar equation is \(r = \frac{1}{4} - \sin \theta\).

Key Concepts

Cartesian Coordinates

The Cartesian coordinate system, also known as the rectangular coordinate system, is fundamental in mathematics to locate points on a plane. It uses two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point on this plane is represented by a pair of numerical coordinates

• The first value \( x \) tells the position along the horizontal direction.
• The second value \( y \) tells the position along the vertical direction.

For example, the equation of a circle \( x^2 + y^2 + y = 0 \) is written in Cartesian coordinates. Here, both the x and y variables must satisfy the equation for any point (x, y) that lies on the circle. This system is intuitive because it directly relates to the concept of coordinates in the Euclidean plane, where distance and angles can be easily calculated using simple algebra.

Polar Coordinates

Polar coordinates offer a different way to describe the position of points in a plane using a distance and an angle. This system is particularly useful for problems involving circular and rotational symmetry. A point in the polar coordinate system is defined by:

• \( r \) is the radial distance from the origin (also known as the pole).
• \( \theta \) is the angular coordinate, which is usually measured in radians and represents the angle from the positive x-axis.

Whereas Cartesian coordinates use (x, y) to describe a point, the polar coordinates use (r, \( \theta \)). For example, in our circle problem, converting the Cartesian equation into a polar form involves using the relations \( x = r \cos \theta \) and \( y = r \sin \theta \). These help to transform points from the xy-plane into a new polar setting.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a form that is easier to work with, typically \( (y + k)^2 \). This process can make the equation more conducive for conversion, such as turning a quadratic in Cartesian form into one that clearly represents a geometric shape like a circle.
Here’s how you complete the square for an equation like \( x^2 + y^2 + y = 0 \):

• First, isolate the quadratic \( y \) component: \( x^2 + (y^2 + y) = 0 \).
• Take half of the \( y \) coefficient, then square it: \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \).
• Add and subtract this square within the equation: \( x^2 + (y+\frac{1}{2})^2 - \frac{1}{4} = 0 \).
• Simplify by moving terms to balance: \( x^2 + (y+\frac{1}{2})^2 = \frac{1}{4} \).

This simplification helps in visualizing the circle's parameters through its square form, emphasizing the center and radius of the circle.

Conversion to Polar Form

Converting a Cartesian equation like \( x^2 + y^2 + y = 0 \) into polar form involves substituting the polar coordinate relations \( x = r \cos \theta \) and \( y = r \sin \theta \) into the equation.
Here's a concise outline of the process:

• Substitute the expressions: \( x^2 + ((r\sin\theta)^2 + r\sin\theta) \).
• The equation adjusts as: \( r^2 = \frac{1}{4} - r\sin\theta \).
• Simplify the equation to utilize the polar coordinates: \( r = \frac{1}{4} - \sin \theta \).

This resulting polar equation describes the same circle in a new form, displaying both the circle’s shifting in space and its inherent symmetry. Polar form can sometimes dramatically simplify visualizing or solving problems involving curves and paths that are circular or radial in nature.