Math Problem Statement
Solution
We need to solve the problem step by step. Here's the setup:
We are tasked to:
- Find the equation of a circle that:
- Has its center on the line ,
- Passes through the points of intersection of the given circles:
Step 1: Find points of intersection of the two circles
The two circles are:
Expanding both equations: For Circle 1: (x^2 - 2x + 1) + (y^2 + 10y + 25) = 50 \implies x^2 + y^2 - 2x + 10y + 26 = 50 \implies x^2 + y^2 - 2x + 10y - 24 = 0. \tag{1}
For Circle 2: (x^2 + 2x + 1) + (y^2 + 2y + 1) = 10 \implies x^2 + y^2 + 2x + 2y + 2 = 10 \implies x^2 + y^2 + 2x + 2y - 8 = 0. \tag{2}
Now, subtract from : -4x + 8y - 16 = 0 \implies -x + 2y - 4 = 0 \implies x = 2y - 4. \tag{3}
Step 2: Substitute into one circle equation
Using in : Expand and simplify: So, or .
If , then . If , then .
Thus, the points of intersection are:
Step 3: Equation of the required circle
The circle passes through and and has its center on the line . Let the center of the circle be , where , so .
The equation of the circle is: Substitute : (x - h)^2 + (y + h)^2 = r^2. \tag{4}
Step 4: Use the given points to find and
Substitute into : (16 + 8h + h^2) + h^2 = r^2 \implies 2h^2 + 8h + 16 = r^2. \tag{5}
Substitute into : h^2 + (4 + 4h + h^2) = r^2 \implies 2h^2 + 4h + 4 = r^2. \tag{6}
From and :
Thus, , and the center is .
Step 5: Find
Substitute into :
Final Answer:
The equation of the required circle is:
Would you like a detailed explanation for any step?
Related Questions:
- How do you find the equation of a circle passing through three points?
- How are the points of intersection of two circles determined geometrically?
- What is the significance of a circle’s center lying on a line?
- How can substitution help in solving simultaneous equations of conic sections?
- What other properties of circles can be derived from their equations?
Tip:
Always simplify the circle equations before solving intersection problems to reduce computational complexity.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Circles
Simultaneous Equations
Algebraic Substitution
Formulas
Equation of a circle: (x - h)^2 + (y - k)^2 = r^2
Expanding squared terms: (x ± a)^2 = x^2 ± 2ax + a^2
Solving simultaneous equations: Substitution and elimination
Theorems
Properties of Circles
Intersection of Two Circles
Suitable Grade Level
Grade 11-12
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