Question

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 $(x_1,y_1)$ and $(x_2,y_2)$, which may be real or non real. Find the value of $(x_1^3 + x_2^3 + y_1^3 + y_2^3)$.

Answer 1

40

Answer 2

1. Represent the family of circles passing through points A and B as $S + \lambda L = 0$, where $L$ is the line AB and $S$ is a circle through A and B.
2. Derive the general equation of a circle orthogonal to this family. This results in two conditions on the coefficients $(g, f, c')$ of the orthogonal circle, relating them to the coefficients of the family.
3. The orthogonal circles form a pencil of circles, whose fixed points are the intersection of $S' = 0$ and $L' = 0$, where $S'$ and $L'$ are derived from the conditions in step 2.
4. Solve the system $S'=0$ and $L'=0$ to find the coordinates of the two fixed points $(x_1, y_1)$ and $(x_2, y_2)$. This leads to a quadratic equation for $x$ (or $y$) and a linear relation between $x$ and $y$.
5. Use Vieta's formulas to find $x_1+x_2$, $x_1x_2$, $y_1+y_2$, and $y_1y_2$.
6. Calculate $x_1^3+x_2^3$ and $y_1^3+y_2^3$ using the identity $a^3+b^3 = (a+b)^3 - 3ab(a+b)$.
7. Sum the results to get the final answer.