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Magazine Chapter 7: Problem 48
Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$x^{2}-12 x+y^{2}+10 y=-25$$
Short Answer
Expert verified
Yes, the graph is a circle with center (6, -5) and radius 6.
Step by step solution
01
Identify the General Form of a Circle Equation
A circle's standard equation is \((x-h)^2 + (y-k)^2 = r^2\). Here, compare the given equation \(x^{2}-12x+y^{2}+10y=-25\) with the general form. Notice how the terms involving \(x\) and \(y\) are quadratic, which is a feature of a circular equation.
02
Rearrange and Complete the Square
Rearrange the equation: \(x^{2}-12x + y^{2}+10y = -25\). We need to complete the square for both \(x\) and \(y\) terms. First, rearrange the \(x\) terms: \([x^2 -12x]\). Complete the square by adding and subtracting \(36\) (\((12/2)^2\)), resulting in \((x-6)^2 - 36\).
03
Complete the Square for y
Perform a similar process for \(y\): rearrange the \(y\) terms: \([y^2 + 10y]\). Complete the square by adding and subtracting \(25\) (\((10/2)^2\)), resulting in \((y+5)^2 - 25\).
04
Rewrite the Equation in Circle Form
Substitute back the completed squares into the equation: \((x-6)^2 - 36 + (y+5)^2 - 25 = -25\). Simplify to obtain the form of a circular equation: \((x - 6)^2 + (y + 5)^2 = 36\), noting that it equals \(r^2 = 36\).
05
Identify the Center and Radius
Now that the equation is in the standard circle form \((x-h)^2 + (y-k)^2 = r^2\), we can identify the center \((h, k)=(6, -5)\) and the radius \(r=\sqrt{36} = 6\).
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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 method used to transform a quadratic expression into a perfect square form. This technique is especially handy when dealing with circles, as it helps rewrite the equation in a recognizable circle form.
To complete the square for a term like \[x^2 + bx\], follow these steps:
To complete the square for a term like \[x^2 + bx\], follow these steps:
- Take half of the coefficient of the x-term, \(b\), and square it. This value is referred to as \((b/2)^2\).
- Add and subtract this squared term within the equation. This doesn't change the equality since adding and subtracting the same number is essentially adding zero.
- Factor the completed perfect square trinomial, which now appears as \((x + b/2)^2\).
Standard Form of a Circle
The standard form of a circle provides a clear representation of a circle's equation, making it easy to decipher important characteristics like the circle's center and radius.
The formula for the standard form of a circle is:\[(x - h)^2 + (y - k)^2 = r^2\]where:
The formula for the standard form of a circle 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.
Equation of a Circle
To determine whether an equation represents a circle on the graph, it is essential to rearrange it into the standard form of a circle equation.
Given an equation such as\[x^2 - 12x + y^2 + 10y = -25,\]one must reorganize and complete the square for both variables, \(x\) and \(y\), to transform the expression.
After completing the square:
Given an equation such as\[x^2 - 12x + y^2 + 10y = -25,\]one must reorganize and complete the square for both variables, \(x\) and \(y\), to transform the expression.
After completing the square:
- The equation takes the form\[(x-6)^2 + (y+5)^2 = 36,\]
Identifying Center and Radius
Identifying the center and radius of a circle from its equation allows you to understand its geometry and plot it accurately.
Given the circle's equation in the standard form\[(x-h)^2 + (y-k)^2 = r^2,\]here's how to find the components:
Given the circle's equation in the standard form\[(x-h)^2 + (y-k)^2 = r^2,\]here's how to find the components:
- The center \( (h, k) \) can be directly identified as the coordinates paired with the variables \(x\) and \(y\) after completing the square.
- The radius \(r\) is derived from the equation's right-hand side, noting that it is the square root of the given number (\(r^2\)).
