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Chapter 11: Problem 7

Find the equation of a circle satisfying the given conditions. Center: \((-4,3) ;\) radius: 2

Short Answer

Expert verified
(x + 4)^2 + (y - 3)^2 = 4

Step by step solution

01

- Understand the General Form of a Circle's Equation

The general form of a circle's equation with center \((h, k)\) and radius \(r\) is given by: \( (x - h)^2 + (y - k)^2 = r^2 \).
02

- Identify the Given Values

Identify the center and radius from the given conditions. Here, the center \( (h, k) \) is \( (-4, 3) \) and the radius \( r \) is 2.
03

- Substitute the Values into the Equation

Substitute \( h = -4 \), \( k = 3 \), and \( r = 2 \) into the general form. This results in the equation: \[ (x + 4)^2 + (y - 3)^2 = 2^2 \].
04

- Simplify the Equation

Simplify the equation by squaring the radius: \[ (x + 4)^2 + (y - 3)^2 = 4 \]. This is the final equation of the circle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

circle equation
In geometry, the equation of a circle is a mathematical representation of all the points that are equidistant from a central point. This equation helps us identify the location and size of the circle. Specifically, the standard form of a circle's equation with center \( (h, k) \) and radius \( r \) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
When learning about circles, it is crucial to understand this formula as it forms the foundation for more advanced concepts and problems.
radius and center of circle
The radius and center are the two most critical properties of any circle. The center of a circle is the point from which all points on the circumference are equally distant. It's represented as \( (h, k) \) in the equation.
The radius, denoted by \( r \), is the distance from the center to any point on the circle's edge. In our exercise, the center was given as \( (-4, 3) \) and the radius was 2.
To substitute these values into the standard form of the circle's equation:
  • Plugging in the center coordinates: \( h = -4 \) and \( k = 3 \)
  • Entering the radius value: \( r = 2 \)
We get the equation: \[ (x + 4)^2 + (y - 3)^2 = 2^2 \]
general form of a circle
The general form of a circle's equation helps simplify problems by providing a clear, standardized way to represent circles. After substituting the specific center and radius values into the standard equation, we get the simplified form:
\[ (x + 4)^2 + (y - 3)^2 = 4 \]
This simplification involves just squaring the radius. In this example, \( 2^2 = 4 \), so we replace \( r^2 \) with 4 in the formula.
Remember that understanding how to transition from the general equation to this simplified form makes solving circle-related problems much more manageable.

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Most popular questions from this chapter

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