Mock AIME 1 2006-2007 Problems/Problem 6

Problem

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 line 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 $a+b+c$ .

Solution

From the condition that $\mathcal L$ is tangent to $P_1$ we have that the system of equations $ax + by = c$ and ${y = x^2 + \frac{101}{100}}$ has exactly one solution, so $ax + b(x^2 + \frac{101}{100}) = c$ has exactly one solution. A quadratic equation with only one solution must have discriminant equal to zero, so we must have $a^2 - 4\cdot b \cdot (\frac{101}{100}b - c) = 0$ or equivalently $25a^2 -101b^2 + 100bc = 0$ . Applying the same process to $P_2$ , we have that $a(y^2 + \frac{45}4) + by = c$ has a unique root so $b^2 - 4\cdot a \cdot (\frac{45}4a - c) = 0$ or equivalently $b^2 - 45a^2 + 4ac = 0$ . We multiply the first of these equations through by $a$ and the second through by $b$ and subtract in order to elliminate $c$ :

$25a^3 - 101ab^2 + 45a^2b - b^3 = 0$ . We know that the slope of $\mathcal L$ , $-\frac b a$ , is a rational number, so we divide this equation through by $-a^3$ and let $\frac b a = q$ to get $q^3 - 45q^2 + 101q - 25 = 0$ .