Math Problem Statement
Determine the equation of the radical axis of the two circles x²+ y²− 4x + 6y − 12 = 0 and 4x²+ 4y²− 8x + 48y − 108 =0
Solution
To find the radical axis of two circles, we subtract the equations of the circles to eliminate the quadratic terms. Here's the step-by-step solution:
Given equations:
Step 1: Normalize the second equation
Divide the second equation by 4 to simplify:
So the two equations to work with are:
Step 2: Subtract from
Simplify:
Step 3: Finalize the equation of the radical axis
Divide through by 3 for simplicity:
Final Answer:
The equation of the radical axis is:
Would you like me to explain the concept of the radical axis in more depth or verify this using another method?
Related Questions:
- What is the geometric significance of the radical axis of two circles?
- How does the radical axis relate to the power of a point with respect to the circles?
- What happens if the two circles intersect? How does the radical axis behave?
- How do we find the point of intersection of the radical axis with one of the circles?
- Can we determine the center and radius of both circles from their equations?
Tip: Always ensure equations of circles are simplified before finding their radical axis to avoid unnecessary complexity!
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Math Problem Analysis
Mathematical Concepts
Analytical Geometry
Circle Equations
Radical Axis
Formulas
Equation of a circle: (x - h)^2 + (y - k)^2 = r^2
Radical axis: Subtracting the equations of two circles to find the locus of points with equal power with respect to the circles
Theorems
Radical Axis Theorem
Suitable Grade Level
Grades 10-12
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