Maximizing in a Real Quadratic

Algebra Level 5

$a, b$ and $c$ are real values satisfying $a + 2b + 3c = 195$ and $a^2 + b^2 + c^2 = 2925$ . The maximum value of $a$ has the form $\frac {p}{q}$ , where $p$ and $q$ are positive, coprime integers. What is the value of $p + q$ ?