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Chapter 7: Problem 1

Find the standard equation of the circle and then graph it. Center \((-1,-5),\) radius 10

Short Answer

Expert verified
The standard equation is \((x + 1)^2 + (y + 5)^2 = 100\) with center \((-1, -5)\) and radius 10.

Step by step solution

01

Identify the Elements

The center of the circle is given as \((-1, -5)\) and the radius is 10.
02

Use the Circle Equation Formula

The standard equation of a circle with center \((h, k)\) and radius r is: \[ (x - h)^2 + (y - k)^2 = r^2 \] In this case, \(h = -1\), \(k = -5\), and \(r = 10\).
03

Substitute into the Equation

Plug the values into the circle equation formula: \[ (x - (-1))^2 + (y - (-5))^2 = 10^2 \]Simplify to get: \[ (x + 1)^2 + (y + 5)^2 = 100 \]
04

Graph the Circle

To graph the circle:1. Plot the center of the circle at \((-1, -5)\).2. Use the radius, which is 10, to mark points 10 units away from the center in all directions to outline the circle.3. Draw the circle ensuring it is smooth and centered on \((-1, -5)\).

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Key Concepts

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

Understanding the Radius
The radius of a circle is the distance from its center to any point on its circumference. It is a crucial element in the geometry of a circle as it helps define the size and scale of the circle.
Understanding the radius is important because it directly impacts both the size and the equation of the circle.
  • The longer the radius, the larger the circle.
  • Conversely, a shorter radius results in a smaller circle.
In our exercise, the radius is given as 10.
This means every point on the circle is 10 units away from the given center (-1, -5). When writing the equation of the circle, the square of the radius ( 10 squared) is used, which in this case equals 100. Understanding this basic property is essential for visualizing and drawing the circle accurately.
Exploring the Circle Equation Formula
The circle equation formula is a fundamental concept in geometry. It allows us to represent circles algebraically using the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \] This equation describes a circle in terms of its center and radius:
  • \( (x - h) \) and \( (y - k) \) denote the shifts in the x and y directions from the circle's center.
  • \( h \) and \( k \) specify the coordinates of the center of the circle.
  • \( r \) represents the radius.
Given the center (-1, -5) and our radius 10, you substitute these values into the formula, leading to \( (x + 1)^2 + (y + 5)^2 = 100 \) after simplification. This serves as the basis for both graphing and further mathematical applications of the circle.
Finding the Center of a Circle
The center of a circle is a key reference point for defining the circle's location in a coordinate system. In our exercise, this is represented by the point (-1, -5).
  • The first value, -1, is the x-coordinate.
  • The second value, -5, is the y-coordinate.
This center dictates where the entire circle will be positioned on the plane. Knowing the center is critical because:
  • It allows you to write the circle's equation accurately, which involves substituting the center's coordinates into \( h \) and \( k \) of the circle equation formula.
  • When graphing, the center is your starting point from which you'll measure the radius outward to complete the drawing.
Locating and using the center correctly ensures that the circle is placed precisely, maintaining its intended geometric properties and size.

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