Math Problem Statement
The general equation for a circle is : a(x^2 + y^2) + bx + cy + d = 0 There is exactly one circle passing through the points (-2,-3), (3,-3), and (0,0). Construct an equation for this circle
Solution
To find the equation of the circle passing through the points , , and , we need to substitute these points into the general equation for a circle:
Step 1: Substitute the points into the equation.
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Substitute : Simplifying:
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Substitute : Simplifying:
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Substitute :
Since , the equation simplifies to:
Step 2: Solve the system of linear equations.
The equations from substituting the points now are:
We solve these simultaneously.
Solve the first equation for :
Substitute into the second equation:
Multiply the terms by 2 to eliminate the fraction:
So, .
Substitute back into :
Step 3: Write the equation of the circle.
Substitute , , and into the general form:
Factor out :
Since , we can divide by to get:
This is the equation of the circle passing through the points , , and .
Would you like more details or have any questions?
Relative Questions:
- How do you find the center and radius of the circle given its equation?
- Can the method used here be applied to find the circle passing through any three points?
- What is the significance of the coefficients and in the general circle equation?
- How does the equation change if the circle is centered at the origin?
- How do you derive the standard form of a circle from the general form?
Tip:
When given three points, ensuring they are not collinear is crucial for a unique circle to exist that passes through them.
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Math Problem Analysis
Mathematical Concepts
Circle Equations
Linear Equations
System of Equations
Formulas
General equation of a circle: a(x^2 + y^2) + bx + cy + d = 0
Theorems
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Suitable Grade Level
Grades 11-12
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