The circles, which cut the family of circles passing through the fixed points A = (2, 1) and B = (4, 3) orthogonally, pass through two fixed points (x1, y₁) and (x2, y2), which may be real or non real. Find the value of (x_{1} ^ 3 + x_{2} ^ 3 + y_{1} ^ 3 + y_{2} ^ 3) .


To solve this problem, we need to analyze the condition of orthogonality between two families of circles, specifically those that intersect a family of circles passing through fixed points \( A = (2, 1) \) and \( B = (4, 3) \).

1. **Equation of the Family of Circles Passing Through \( A \) and \( B \):**
   Any circle passing through two fixed points \( A \) and \( B \) can be represented as:
   \[
   S = 0 \Rightarrow (x - 2)^2 + (y - 1)^2 + \lambda \left[(x - 4)^2 + (y - 3)^2 \right] = 0
   \]
   where \( \lambda \) is a parameter.

2. **Orthogonality Condition:**
   For a circle to intersect another family of circles orthogonally, the sum of their gradients at any intersection point must be zero. The condition for orthogonality between two circles with equations \( S = 0 \) and \( T = 0 \) is given by:
   \[
   2g g' + 2f f' = c + c'
   \]
   where \( g, f, \) and \( c \) are the coefficients in the general form of a circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \).

3. **Determine the Fixed Points \( (x_1, y_1) \) and \( (x_2, y_2) \):**
   Since the circles are orthogonal to the family passing through \( A \) and \( B \), they must pass through the points where this family intersects the line at infinity, leading us to a fixed pair of points, which can often include complex values if no real intersection exists.

4. **Calculate \( (x_1^3 + x_2^3 + y_1^3 + y_2^3) \):**
   The values of \( x_1, x_2, y_1, \) and \( y_2 \) would be determined by solving the orthogonal condition explicitly, which may reveal real or complex roots.

Given these conditions, a detailed solution would involve either directly computing the values of \( x_1, x_2, y_1, \) and \( y_2 \) or deducing them based on symmetry properties.

The circles, which cut the family of circles passing through the fixed points A = (2, 1) and B = (4, 3) orthogonally, pass through two fixed points (x1, y₁) and (x2, y2), which may be real or non real. Find the value of (x_{1} ^ 3 + x_{2} ^ 3 + y_{1} ^ 3 + y_{2} ^ 3).

Solve a circle geometry problem involving orthogonal circles and fixed points through coordinates A = (2, 1) and B = (4, 3). Learn how to determine the value of (x₁^3 + x₂^3 + y₁^3 + y₂^3) using orthogonality conditions and circle equations. Suitable for advanced geometry students in Grades 11-12.

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