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Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 6"

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==Solution==
 
==Solution==
  
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From the condition that <math>\mathcal L</math> is tangent to <math>P_1</math> we have that the system of equations <math>ax + by = c</math> and <math> {y = x^2 + \frac{101}{100}}</math> has exactly one solution, so <math>ax + b(x^2 + \frac{101}{100}) = c</math> has exactly one solution.  A [[quadratic equation]] with only one solution must have [[discriminant]] equal to [[zero (constant) | zero]], so we must have <math>a^2 - 4\cdot b \cdot (\frac{101}{100}b - c) = 0</math> or equivalently <math>25a^2 -101b^2 + 100bc = 0</math>.  Applying the same process to <math>P_2</math>, we have that <math>a(y^2 + \frac{45}4) + by = c</math> has a unique root so <math>b^2 - 4\cdot a \cdot (\frac{45}4a - c) = 0</math> or equivalently <math>b^2 - 45a^2 + 4ac = 0</math>.  We multiply the first of these equations through by <math>a</math> and the second through by <math>b</math> and subtract in order to elliminate <math>c</math>:
+
From the condition that <math>\mathcal L</math> is tangent to <math>P_1</math> we have that the system of equations <math>ax + by = c</math> and <math> {y = x^2 + \frac{101}{100}}</math> has exactly one solution, so <math>ax + b(x^2 + \frac{101}{100}) = c</math> has exactly one solution.  A [[quadratic equation]] with only one solution must have [[discriminant]] equal to [[zero (constant) | zero]], so we must have <math>a^2 - 4\cdot b \cdot (\frac{101}{100}b - c) = 0</math> or equivalently <math>25a^2 -101b^2 + 100bc = 0</math>.  Applying the same process to <math>P_2</math>, we have that <math>a(y^2 + \frac{45}4) + by = c</math> has a unique root so <math>b^2 - 4\cdot a \cdot (\frac{45}4a - c) = 0</math> or equivalently <math>b^2 - 45a^2 + 4ac = 0</math>.  We multiply the first of these equations through by <math>a</math> and the second through by <math>25b</math> and subtract in order to elliminate <math>c</math>:
  
−
<math>25a^3 - 101ab^2 + 45a^2b - b^3 = 0</math>.  We know that the slope of <math>\mathcal L</math>, <math>-\frac b a</math>, is a rational number, so we divide this equation through by <math>-a^3</math> and let <math>\frac b a = q</math> to get <math>q^3 - 45q^2 + 101q - 25 = 0</math>.
+
<math>25a^3 - 101ab^2 + 1125a^2b - 25b^3 = 0</math>.  We know that the slope of <math>\mathcal L</math>, <math>-\frac b a</math>, is a rational number, so we divide this equation through by <math>-a^3</math> and let <math>\frac b a = q</math> to get <math>25q^3 - 1125q^2 + 101q - 25 = 0</math>.
  
 
{{solution}}
 
{{solution}}

Revision as of 18:50, 29 September 2006

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 $25b$ and subtract in order to elliminate $c$:

$25a^3 - 101ab^2 + 1125a^2b - 25b^3 = 0$. 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 $25q^3 - 1125q^2 + 101q - 25 = 0$.

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