Problem 32 - Algebra 2

Question

Write an equation for the circle that satisfies each set of conditions. center \((8,-9),\) passes through \((21,22)\)

Step-by-Step Solution

Answer

The equation is \\( (x - 8)^2 + (y + 9)^2 = 1130 \\\).

Step 1: Identify the Circle Equation

The general equation for a circle with center \( (h, k) \) is given by \( (x - h)^2 + (y - k)^2 = r^2 \), where \( r \) is the radius.

Step 2: Plug in the Center Coordinates

Substitute the center \( (8, -9) \) into the equation to get \( (x - 8)^2 + (y + 9)^2 = r^2 \).

Step 3: Use the Point to Find the Radius

Substitute the point \( (21, 22) \) into the equation to solve for the radius \( r \). This gives \[ (21 - 8)^2 + (22 + 9)^2 = r^2 \] \[ 13^2 + 31^2 = r^2 \] \[ 169 + 961 = r^2 \] \[ 1130 = r^2 \].

Step 4: Write the Final Equation

Now that we know \( r^2 = 1130 \), substitute back into the circle's equation to get \( (x - 8)^2 + (y + 9)^2 = 1130 \).

Key Concepts

Radius Calculation

To calculate the radius of a circle, you need to know the center coordinates and a point on the circle. In our exercise, the circle's center is at \( (8, -9) \) and it passes through the point \( (21, 22) \). To find the radius, use the distance formula between these two points. This distance is the radius.

The distance (or radius, \( r \)) is computed using the following steps:

• Subtract the x-coordinates: \( 21 - 8 = 13 \).
• Subtract the y-coordinates: \( 22 - (-9) = 31 \).
• Use the Pythagorean theorem: \( r = \sqrt{(13)^2 + (31)^2} \).
• Calculate: \( r = \sqrt{169 + 961} = \sqrt{1130} \).

Now, \( \sqrt{1130} \) is the radius of the circle. In a circle equation, you typically use \( r^2 \), which is 1130. Understanding how to calculate the radius is crucial for writing the circle's equation.

Center Coordinates

Determining the center coordinates is essential for forming the circle equation. The center of a circle is represented as \( (h, k) \). For this exercise, the center coordinates are \( (8, -9) \).

These coordinates serve as a fixed point from which every point on the circle is equidistant. To ensure the circle equation captures this relationship, substitute the center into the general circle equation:

• Start with the general form: \( (x - h)^2 + (y - k)^2 = r^2 \).
• Substitute \( h = 8 \) and \( k = -9 \) to get: \( (x - 8)^2 + (y + 9)^2 = r^2 \).

Correctly identifying and using the center coordinates ensures that the resulting circle equation accurately represents the given circle.

Point on Circle

In any circle problem, knowing a point on the circle helps to determine the circle's radius. A point on the circle directly relates to the radius calculation, as shown in our exercise.

For a point \( (x_1, y_1) \) on a circle with a known center \( (h, k) \), the radius is found by making:

• \( (x_1 - h)^2 + (y_1 - k)^2 = r^2 \) true.
• In our case: \( (21 - 8)^2 + (22 + 9)^2 = r^2 \) confirms that \( 13^2 + 31^2 = 1130 \), validating our radius calculation previously described.

Using a point on the circle not only confirms the circle's radius but also assures the accuracy of your circle equation. This information fundamentally ties into ensuring every element in the circle's definition is accurate.

Circle Formula

The circle formula is pivotal to any problem involving circles. The standard form equation is:

• \( (x - h)^2 + (y - k)^2 = r^2 \)

This formula describes all the points \( (x, y) \) that constitute the circle centered at \( (h, k) \) with radius \( r \).

In the exercise, we used this formula to write a specific circle equation. By substituting the center \( (8, -9) \) and the calculated \( r^2 = 1130 \), we get:

• The circle equation \( (x - 8)^2 + (y + 9)^2 = 1130 \).

Mastering this formula allows you to transition from abstract circle concepts to precise definitions capable of solving real problems. Remember, this consistent approach will provide clarity and aid in solving a wide array of geometric problems efficiently.