Question

Write the standard form of the equation and the general form of the equation of each circle of radius \(r\) and center \((h, k)\). Graph each circle. $$ r=3 ;(h, k)=(1,0) $$

Short answer

Standard form: \((x - 1)^2 + y^2 = 9\); General form: \(x^2 + y^2 - 2x - 8 = 0\).

Step-by-step solution

Identify the given information

Given the radius of the circle, \(r = 3\), and the center of the circle, \((h, k) = (1, 0)\).

Write the standard form of the circle's equation

The standard form of a circle's equation with radius \(r\) and center \((h, k)\) is \((x - h)^2 + (y - k)^2 = r^2\). Substitute the given values into this formula:ewline \((x - 1)^2 + (y - 0)^2 = 3^2\).ewline Simplify it to get the standard form:ewline \((x - 1)^2 + y^2 = 9\).

Expand and simplify to get the general form

Expand \((x - 1)^2 + y^2 = 9\):ewline \(x^2 - 2x + 1 + y^2 = 9\).ewline Combine like terms and move all terms to one side of the equation to get the general form:ewline \(x^2 + y^2 - 2x + 1 - 9 = 0\)ewline Simplify:ewline \(x^2 + y^2 - 2x - 8 = 0\).

Graph the circle

Plot the center of the circle at \((1, 0)\). Draw a circle with radius 3 units around this center.

Key concepts

Standard Form of the Equation of a Circle

Understanding the standard form of a circle's equation is crucial as it allows easy identification of a circle's center and radius. The standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \). Here, \( (h, k) \) represents the center of the circle, and \( r \) represents the radius. This form is particularly useful for graphing and quickly visualizing the circle.
Given the example where \( r = 3 \) and the center \( (h, k) = (1, 0) \), substitute these values into the formula to get:

• \( (x - 1)^2 + (y - 0)^2 = 3^2 \)

This simplifies to: \( (x - 1)^2 + y^2 = 9 \)
By understanding and using this standard form, you can easily identify and use key features of the circle, making it easier to solve related math problems.

General Form of the Equation of a Circle

The general form of a circle's equation is another way to express the circle mathematically, which involves expanding and simplifying the standard form. The general form is written as: \( Ax^2 + By^2 + Cx + Dy + E = 0 \), where \( A \) and \( B \) are typically 1.
Let's take the standard form of our given circle and convert it into general form:

• Standard form: \( (x - 1)^2 + y^2 = 9 \)
• Expand: \( x^2 - 2x + 1 + y^2 = 9 \)
• Combine like terms: \( x^2 + y^2 - 2x + 1 - 9 = 0 \)
• Simplify: \( x^2 + y^2 - 2x + 8 = 0 \)

The advantage of the general form is its compatibility with various algebraic methods and the fact that it integrates into broader linear equation systems. However, it might be less intuitive for immediate graphical interpretation compared to the standard form.

Graphing Circles

Graphing circles involves understanding and applying both the standard and general forms of the equation. First, you determine the center and radius from the standard form of the circle's equation.
From our example:

• Radius (\( r \)): 3
• Center (\( h, k \)): \( (1, 0) \)

To graph this circle:

• Plot the center at \( (1, 0) \)
• From the center, use the radius to mark points 3 units away in all directions.
• Draw the circle by connecting these points, ensuring the circle is symmetrical around the center.

This visualization helps in verifying solutions and understanding the geometric properties of circles better. Graphing creates a visual context, making the abstract concept of algebraic equations more tangible.
Remember, always double-check for symmetry and the correct radius to ensure accuracy.

Radius and Center

The radius and center are fundamental aspects of a circle. In the context of equations, identifying these from the equation can simplify various tasks, like graphing or converting equations between forms.
The center, represented by \( (h, k) \), is the fixed point from which every point on the circle is equidistant. This equidistance is known as the radius (\( r \)).
For example, in the given problem:

• Center: \( (1, 0) \)
• Radius: \( 3 \)

To find the center and radius from the standard form equation \( (x - h)^2 + (y - k)^2 = r^2 \), look at \( h \) and \( k \) for the center and \( \sqrt{r^2} \) for the radius.
This information is crucial for transforming equations, particularly when going from standard to general form or vice versa. Understanding these properties enhances your ability to manipulate and accurately interpret various forms of a circle's equation.