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Magazine Chapter 0: Problem 56
Find the center and radius of each circle. $$x^{2}+y^{2}+8 x+2 y-28=0$$
Short Answer
Expert verified
The center is \((-4, -1)\) and the radius is \(\sqrt{45}\).
Step by step solution
01
Group terms
First, rearrange the given equation by grouping the x and y terms together as follows: \[ x^2 + 8x + y^2 + 2y = 28 \] The right side of the equation is the constant term moved from the left.
02
Complete the square for x
To complete the square on the x terms, take the coefficient of x, which is 8, halve it to get 4, and then square it to get 16. Add and subtract 16 to maintain equality:\[ x^2 + 8x = (x+4)^2 - 16 \] Thus, our equation becomes: \[ (x+4)^2 - 16 + y^2 + 2y = 28 \]
03
Complete the square for y
To complete the square on the y terms, take the coefficient of y, which is 2, halve it to get 1, then square it to obtain 1. Add and subtract 1:\[ y^2 + 2y = (y+1)^2 - 1 \] Now, include this in the equation:\[ (x+4)^2 - 16 + (y+1)^2 - 1 = 28 \]
04
Simplify the equation
Combine all the constants:\[ (x+4)^2 + (y+1)^2 - 17 = 28 \]Add 17 to both sides:\[ (x+4)^2 + (y+1)^2 = 45 \]
05
Identify center and radius
The equation \[(x+4)^2 + (y+1)^2 = 45\] 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, the center \( (h, k) \) is \((-4, -1)\) and the radius \( r \) is \( \sqrt{45} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a crucial algebraic technique that helps in rewriting quadratic expressions into perfect square form. This is especially useful in the context of circle equations. Generally, if you have a quadratic expression like \(x^2 + bx\), you aim to reformat it into \((x + d)^2 - d^2\), where \(d\) is half of \(b\). To see how this works, let's break it down step by step:
- Start with the expression \(x^2 + bx\).
- Take the coefficient of \(x\), which is \(b\), divide it by 2, and then square the result to find \(d^2\). This gives \(d = \frac{b}{2}\).
- Add and subtract \(d^2\) to complete the square: \(x^2 + bx = (x + d)^2 - d^2\).
- Now, rearrange or add these terms to your equation as needed.
Center of a Circle
The center of a circle is an essential component that helps to fully define its position in a coordinate plane. When given the standard equation of a circle, \((x-h)^2 + (y-k)^2 = r^2\), the center is represented by the point \((h, k)\). Understanding the center allows us to precisely locate where the circle is situated.
Here's how the center is determined:
Here's how the center is determined:
- The terms \((x-h)\) and \((y-k)\) in the circle's equation highlight the shifts from the origin point \((0,0)\).
- If \(h = 0\) and \(k = 0\), then the circle is centered exactly at the origin.
- In the example equation we transformed, \((x+4)^2 + (y+1)^2 = 45\), the center \((h, k)\) is \((-4, -1)\). Notice that the signs change from \(+\) in the equation to \(-\) in the center coordinates. This is a key point to remember when identifying the circle's center from its equation.
Radius of a Circle
The radius is another fundamental property of a circle, representing the distance from its center to any point on its perimeter. In the standard form of a circle's equation, \((x-h)^2 + (y-k)^2 = r^2\), the radius \(r\) can be extracted from \(r^2\). Here's how to determine the radius correctly and effectively:
- From the equation \(r^2\), take the square root to calculate the actual radius \(r\).
- In our example, the equation simplifies to \((x+4)^2 + (y+1)^2 = 45\). Hence, \(r^2 = 45\), which means \(r = \sqrt{45}\).
- You can simplify \(\sqrt{45}\) further by recognizing it as \(\sqrt{9 \times 5}\), which equals \(3\sqrt{5}\).
