If a, b and c are non-zero reals such that a + b + c = 11 and \frac {1}{a} + \frac {1}{b} +\frac {1}{c} = 0, what is the value of a^2 + b^2 + c^2?
To find the value of \(a^2 + b^2 + c^2\), we can use the following algebraic manipulation:
\[
\begin{align*}
a + b + c &= 11 \\
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} &= 0
\end{align*}
\]
Let's eliminate the fractions by taking the reciprocals of both sides:
\[
\begin{align*}
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} &= \frac{a+b+c}{abc} \\
&= 0
\end{align*}
\]
Since we know that \(a + b + c = 11\), we can substitute it in:
\[
\frac{11}{abc} = 0
\]
Since the sum of the non-zero real numbers cannot be equal to zero, we conclude that \(abc = 0\). Therefore, at least one of the variables \(a\), \(b\), or \(c\) must be zero.
Now, let's consider the possible cases:
Case 1: \(a = 0\)
If \(a = 0\), we have \(b + c = 11\) since \(a\) is zero. From the equation \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0\), we can conclude that \(\frac{1}{b} + \frac{1}{c} = 0\). This implies that \(\frac{1}{b} = -\frac{1}{c}\), so \(b = -c\).
Substituting \(b = -c\) into \(b + c = 11\), we get \(-c + c = 11\), which is not possible. Hence, \(a = 0\) is not a valid case.
Case 2: \(b = 0\)
In this case, \(a + c = 11\) since \(b\) is zero. From the equation \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0\), we have \(\frac{1}{a} + \frac{1}{c} = 0\). Rearranging this equation, we get \(\frac{1}{a} = -\frac{1}{c}\), so \(a = -c\).
Substituting \(a = -c\) into \(a + c = 11\), we obtain \(-c + c = 11\), which is again not possible. Thus, \(b = 0\) is not a valid case.
Case 3: \(c = 0\)
Similarly, if \(c = 0\), we have \(a + b = 11\) since \(c\) is zero. From the equation \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0\), we have \(\frac{1}{a} + \frac{1}{b} = 0\).
Multiplying both sides by \(ab\), we get \(a + b = 0\). This implies that \(a = -b\).
Substituting \(a = -b\) into \(a + b = 11\), we get \(-b + b = 11\), which gives \(0 = 11\), which is not possible. Therefore, \(c = 0\) is also not a valid case.
From the analysis of all possible cases, we can conclude that there is no non-zero real solution for the given equations. Therefore, there is no valid value for \(a^2 + b^2 + c^2\).