Mock AIME 1 2006-2007 Problems/Problem 6

6. Let $P_{1}: y=x^{2}+\frac{101}{100}$ and $P_{2}: x=y^{2}+\frac{45}{4}$ be two parabolas in the cartesian plane. Let $\mathcal{L}$ be the common tangent of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$ . Find $a+b+c$ .