Problem 86 - Algebra and Trigonometry

Question

Show that the graph of the equation \(r=-2 a \sin \theta, a>0\) is a circle of radius \(a\) with center \((0,-a)\) in rectangular coordinates.

Step-by-Step Solution

Answer

The graph is a circle with radius \(a\) and center \((0, -a)\).

Step 1: Convert polar equation to rectangular equation

To show the graph of the given polar equation \( r = -2a \sin \theta, \ a > 0 \), we need to convert it to rectangular form. We know the conversions: \(\sin \theta = \frac{y}{r}\) and \(r = \sqrt{x^2 + y^2}\).

Step 2: Substitute and simplify

Substitute \( \sin \theta = \frac{y}{r} \) into the given equation:\[-2a \sin \theta = -2a \frac{y}{r}\].Given \(r = \sqrt{x^2 + y^2}\), our equation now looks like:\[ r = -2a \frac{y}{r}\].

Step 3: Multiply both sides by r

Multiply both sides of the equation by \(r\) to eliminate the denominator: \[ r^2 = -2ay\].Since \(r^2 = x^2 + y^2\), substitute it in:\[x^2 + y^2 = -2ay\].

Step 4: Complete the square

To better understand the equation as a circle, complete the square:\[x^2 + y^2 + 2ay = 0\]\[x^2 + (y^2 + 2ay + a^2) = a^2\]Now, we have: \[x^2 + (y + a)^2 = a^2\].

Step 5: Identify the circle

The equation \[x^2 + (y+a)^2 = a^2\] is the standard form of a circle with center at \((0, -a)\) and radius \(a\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using the distance from the origin and the angle from the positive x-axis. They are different from rectangular coordinates, which use x and y coordinates to specify the location of a point. The polar coordinate system is especially useful for dealing with problems involving circles and other round shapes. In polar coordinates, a point is represented as \( (r, \theta) \) where \( r \) is the distance from the origin and \( \theta \) is the angle from the x-axis.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common coordinate system used in mathematics. In this system, each point is defined by two numbers \( (x, y) \). These coordinates represent the point's horizontal and vertical displacement from a reference point known as the origin. One of the key advantages of rectangular coordinates is that they make it easy to work with geometric shapes and algebraic equations. When converting from polar to rectangular coordinates, the following formulas are used:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

This allows you to transform equations and points from the polar form to a rectangular form to make calculations and graphing simpler.

Circle Equations

Equations of circles can be easily represented in both polar and rectangular coordinates. In rectangular coordinates, a standard equation of a circle looks like:

• \( (x - h)^2 + (y - k)^2 = r^2 \)

Here, \( (h, k) \) is the center of the circle and \( r \) represents the radius. The process of converting a polar equation like \( r = -2a \sin \theta \) into a rectangular form helps us identify whether it represents a circle and find its key parameters like its center and radius. By completing the square, you can express the equation in the standard form of a circle equation and easily identify the center and radius from it.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial. This helps in easily identifying the shape and location of geometric figures, especially circles. Here's how you can do it:
Let's say you have an equation involving \( x \) and \( y \), such as \( x^2 + y^2 + 2ay = 0 \). To complete the square:

• Group the \( y \) terms: \( x^2 + (y^2 + 2ay) = 0 \).
• Add and subtract the square of half the coefficient of \( y \): \( x^2 + (y^2 + 2ay + a^2 - a^2) = 0 \).
• Reorganize it into a perfect square trinomial: \( x^2 + (y + a)^2 - a^2 = 0 \).

This enables you to rewrite the equation in the standard form of a circle, making it easy to figure out the center \( (0, -a) \) and radius \( a \).