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Chapter 1: Problem 43

Write the standard equation for each circle. Center at \((-2,5)\) with radius \(1 / 2\)

Short Answer

Expert verified
(x + 2)^2 + (y - 5)^2 = 1/4

Step by step solution

01

- Recall the Standard Equation of a Circle

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

- Identify the Center Coordinates

The center of the circle is given as \((-2, 5)\), so \(h = -2\) and \(k = 5\).
03

- Determine the Radius

The radius of the circle is given as \( \frac{1}{2} \), so \(r = \frac{1}{2} \).
04

- Substitute into the Standard Equation

Substitute \(h = -2\), \(k = 5\), and \(r = \frac{1}{2}\) into the standard equation: \((x + 2)^2 + (y - 5)^2 = \left( \frac{1}{2} \right)^2\).
05

- Simplify

Simplify the right-hand side of the equation: \[ \left( \frac{1}{2} \right)^2 = \frac{1}{4} \]. The equation becomes: \((x + 2)^2 + (y - 5)^2 = \frac{1}{4}\).

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Key Concepts

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

Standard Form of a Circle's Equation
Understanding the standard form of a circle's equation is essential in geometry. The standard equation for a circle is derived when the center of the circle is located at coordinates \( (h, k) \) and the radius is \( r \). This equation is written as:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, each component has a specific role:
- \( (x - h) \) and \( (y - k) \) represent the horizontal and vertical shifts of the circle from the origin.
- Squaring these expressions ensures that the values are nonnegative distances.
- \( r^2 \) represents the square of the circle's radius.
By understanding this formula, you can easily grasp the nature of any circle given its center and radius.
Circle's Center
The circle's center is a critical component of its equation. For a circle centered at \( (-2, 5) \), we use the variables \( h \) and \( k \) to represent these coordinates.
In this case, \( h = -2 \) and \( k = 5 \). Substituting these values into the standard equation, we get:
\[ (x + 2)^2 + (y - 5)^2 = r^2 \]
Always remember that the signs inside the parentheses are opposite to the signs in the coordinates of the center. This is due to the equation's structure where negative signs help to shift the circle to its correct position on the graph.
Radius of the Circle
Finally, let's discuss the radius ( \( r \) ) of the circle. The radius is the distance from the center to any point on the circle. For this exercise, the radius given is \( \frac{1}{2} \).
Inserting the radius into our circle's equation, we need to square it to match the form:
\[ r^2 = \frac{1}{4} \]
Now, substituting this value into the standard equation results in:
\[ (x + 2)^2 + (y - 5)^2 = \frac{1}{4} \]
This final expression fully describes our circle with center \( (-2, 5) \) and radius \( \frac{1}{2} \). By understanding the concept of radius and how to square it, you can accurately complete the standard form of the circle's equation.

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