Problem 73 - Algebra 2

Question

Write an equation of a circle with the given center and radius. center \((3,7),\) radius 6.5

Step-by-Step Solution

Answer

The equation of the circle is \( (x - 3)^2 + (y - 7)^2 = 42.25 \)

Step 1: Identify the center and radius

The center of the circle is given as \((3,7)\), therefore \(h = 3\) and \(k = 7\). The radius is given as 6.5, therefore \(r = 6.5\).

Step 2: Substitute the values

The equation of the circle becomes \( (x - 3)^2 + (y - 7)^2 = 6.5^2 \) .

Step 3: Calculate the square of the radius

The square of 6.5 is 42.25, so the equation of the circle becomes \( (x - 3)^2 + (y - 7)^2 = 42.25 \) .

Key Concepts

Center of a Circle

When working with circles, identifying the center is crucial. Every circle has a specific point at its center, given as coordinates \(h, k\). For our example, the center is provided as \(3, 7\), which means:

• h is the x-coordinate: 3
• k is the y-coordinate: 7

The center directly influences the position of the circle in the coordinate plane. By understanding the center, we can shift the circle left, right, up, or down, without altering its shape or size. The center is our starting point for defining the circle's equation in the standard form.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. For this circle, the radius is given as 6.5. Think of the radius as a tool to understand how wide the circle is from its center across.

• The radius helps in calculating the area and circumference.
• In the circle equation, the radius appears squared.

So, for our exercise, to find the equation, we square the radius: \(6.5^2 = 42.25\). This squared radius is essential in the formulation of the circle's equation.

Circle Equation Standard Form

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) represents the center and \(r\) is the radius. This format helps express the relationship between any point \(x, y\) on the circle and its center.

• Subtract the center coordinates from \(x\) and \(y\).
• Square those differences to track any deviation from the center.
• Equate it to the squared radius for balance.

Plugging in our values, we have: \( (x - 3)^2 + (y - 7)^2 = 42.25 \). This equation precisely represents our circle on the coordinate plane by maintaining its fixed center and consistent radius.