Problem 91 - Algebra and Trigonometry

Question

Find the area of the region that lies outside the circle \(x^{2}+y^{2}=4\) but inside the circle $$ x^{2}+y^{2}-4 y-12=0 $$

Step-by-Step Solution

Answer

The area is \(12\pi\).

Step 1: Identify given equations

We are working with two circles: the first equation is \( x^2 + y^2 = 4 \) and the second equation is \( x^2 + y^2 - 4y - 12 = 0 \). The first equation represents a circle centered at the origin with a radius of 2.

Step 2: Transform the second equation

Rearrange the second equation to make it easier to identify the properties of the circle. Complete the square for the \(y\)-terms in the equation \( x^2 + y^2 - 4y - 12 = 0 \). First, add and subtract \(4\) (\((4/2)^2\)) inside the equation: \( x^2 + (y^2 - 4y + 4) = 12 + 4 \). This simplifies to \( x^2 + (y-2)^2 = 16 \). This represents a circle centered at \((0, 2)\) with a radius of 4.

Step 3: Find regions of interest

Our goal is to find the area that lies outside the smaller circle \( x^2 + y^2 = 4 \), but inside the larger circle \( x^2 + (y-2)^2 = 16 \). The overall area of the larger circle is calculated first, followed by calculating the area of the smaller circle.

Step 4: Calculate area of larger circle

The radius of the larger circle is 4, so the area is given by \(\pi r^2\). Calculate \(\pi(4)^2 = 16\pi\).

Step 5: Calculate area of smaller circle

The radius of the smaller circle is 2, so the area is given by \(\pi r^2\). Calculate \(\pi(2)^2 = 4\pi\).

Step 6: Find desired area

To find the area outside the smaller circle but inside the larger circle, subtract the area of the smaller circle from the area of the larger circle: \(16\pi - 4\pi = 12\pi\).

Key Concepts

Circle equations

In the beautiful world of geometry, circle equations play a significant role in helping us understand the properties of circles. The standard form of a circle's equation is given by the formula \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle and the center is at the origin \((0,0)\). This is exactly the format of the equation \( x^2 + y^2 = 4 \). Here, the circle is centered at the origin with a radius of 2.
Another common form you might see is the circle equation expanded into \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) becomes the center, and \( r \) is the radius. In the event of an equation such as \( x^2 + y^2 - 4y - 12 = 0 \), it needs modification to reveal these parameters, which involves completing the square. Through this, the equation transforms into \( x^2 + (y-2)^2 = 16 \), a circle centered at \((0, 2)\) with a radius of 4.
Even slight shifts in these values significantly impact a circle’s size and position on the coordinate plane. Understanding these transformations helps in visualizing how different circles relate to one another in a given space.

Area calculations

Calculating the area of a circle might seem straightforward, but it’s a fundamental skill in coordinate geometry. The formula \( \pi r^2 \) is your best friend here. It tells us that the area of any circle grows with the square of its radius.
For example, if a circle has a radius of 2, as in the equation \( x^2 + y^2 = 4 \), the area calculation goes like this: \( \pi (2)^2 = 4\pi \). For a larger circle with a radius of 4, as seen in the equation \( x^2 + (y-2)^2 = 16 \), the area becomes \( \pi (4)^2 = 16\pi \).
When finding a region’s area that is bounded by two circles, subtraction becomes important. You’d subtract the smaller circle's area from the larger to determine the area of the region of interest. Thus, the region outside the smaller circle but inside the larger circle here is \( 16\pi - 4\pi = 12\pi \).
It’s a great reminder of how subtraction of areas can reveal facets of geometrical regions.

Completing the square

Completing the square is a nifty algebraic tool that often becomes essential when working with circles. It rearranges an equation to facilitate easier analysis and understanding. This technique is often used to identify the circle's center and radius in non-standard forms.
Consider the equation \( x^2 + y^2 - 4y - 12 = 0 \). By completing the square for the \( y \)-terms, we first focus on these terms alone: \( y^2 - 4y \).

Steps for completing the square:

• Take the coefficient of \( y \), which is -4, divide by 2, and then square it: \((-4/2)^2 = 4\).
• Add and subtract this value inside the equation to make it complete: \( x^2 + (y^2 - 4y + 4) = 16 \).

The adjustment results in a neatly rewritten equation: \( x^2 + (y-2)^2 = 16 \).
Thanks to this procedure, we can easily identify the circle's center \((0, 2)\) and determine its radius, which is 4. Mastering this skill greatly aids in solving complex problems involving circles and their equations.