Math Problem Statement
Solution
To find the equation of the circle, let's first gather the necessary information from the diagram.
Step 1: The general equation of a circle
The equation of a circle with center and radius is given by:
Where:
- is the center of the circle.
- is the radius.
Step 2: Determining the center
From the diagram, it appears the circle is centered at . This is determined by the points where the circle intersects the axes. The x- and y-intercepts of the circle are given as , , , and . The point at looks like the center.
Step 3: Determining the radius
The radius is the distance from the center to any point on the circle. Let's use the point , which is on the circle, and calculate the distance to the center :
Substitute the coordinates and :
Step 4: Equation of the circle
Now that we know the center is and the radius is , we can write the equation of the circle:
Simplifying:
Thus, the equation of the circle is:
Would you like further clarification or details on any part of the solution? Here are some related questions for deeper understanding:
- How do you calculate the distance between two points on the coordinate plane?
- How would you find the radius if only one intercept point and the center are given?
- What if the center of the circle was located on the origin? How would the equation change?
- How can you apply the distance formula in other geometric contexts, such as in triangles or polygons?
- What is the significance of the center and radius in relation to the graph of the circle?
Tip: When solving for the equation of a circle, always double-check that you have correctly identified the center and radius, as these are key to writing the equation.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Equation of a Circle
Distance Formula
Formulas
Equation of a circle: (x - h)^2 + (y - k)^2 = r^2
Distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2)
Theorems
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Suitable Grade Level
Grades 9-12