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:

  1. x2+y24x+6y12=0x^2 + y^2 - 4x + 6y - 12 = 0
  2. 4x2+4y28x+48y108=04x^2 + 4y^2 - 8x + 48y - 108 = 0

Step 1: Normalize the second equation

Divide the second equation by 4 to simplify: x2+y22x+12y27=0x^2 + y^2 - 2x + 12y - 27 = 0

So the two equations to work with are: C1:x2+y24x+6y12=0C_1: x^2 + y^2 - 4x + 6y - 12 = 0 C2:x2+y22x+12y27=0C_2: x^2 + y^2 - 2x + 12y - 27 = 0

Step 2: Subtract C1C_1 from C2C_2

(x2+y22x+12y27)(x2+y24x+6y12)=0(x^2 + y^2 - 2x + 12y - 27) - (x^2 + y^2 - 4x + 6y - 12) = 0 Simplify: 2x+12y27+4x6y+12=0-2x + 12y - 27 + 4x - 6y + 12 = 0 2x+6y15=02x + 6y - 15 = 0

Step 3: Finalize the equation of the radical axis

Divide through by 3 for simplicity: x+3y5=0x + 3y - 5 = 0

Final Answer:

The equation of the radical axis is: x+3y5=0\boxed{x + 3y - 5 = 0}

Would you like me to explain the concept of the radical axis in more depth or verify this using another method?


Related Questions:

  1. What is the geometric significance of the radical axis of two circles?
  2. How does the radical axis relate to the power of a point with respect to the circles?
  3. What happens if the two circles intersect? How does the radical axis behave?
  4. How do we find the point of intersection of the radical axis with one of the circles?
  5. 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