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Difference between revisions of "2002 AIME II Problems/Problem 15"

(Took me a couple years, but I solved it.)
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Circles <math>\mathcal{C}_{1}</math> and <math>\mathcal{C}_{2}</math> intersect at two points, one of which is <math>(9,6)</math>, and the product of the radii is <math>68</math>. The x-axis and the line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be written in the form <math>a\sqrt {b}/c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>.
 
Circles <math>\mathcal{C}_{1}</math> and <math>\mathcal{C}_{2}</math> intersect at two points, one of which is <math>(9,6)</math>, and the product of the radii is <math>68</math>. The x-axis and the line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be written in the form <math>a\sqrt {b}/c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>.
  
−
== Solution ==
+
== Solution 1==
 
Let the smaller angle between the <math>x</math>-axis and the line <math>y=mx</math> be <math>\theta</math>. Note that the centers of the two circles lie on the angle bisector of the angle between the <math>x</math>-axis and the line <math>y=mx</math>. Also note that if <math>(x,y)</math> is on said angle bisector, we have that <math>\frac{y}{x}=\tan{\frac{\theta}{2}}</math>.  Let <math>\tan{\frac{\theta}{2}}=m_1</math>, for convenience. Therefore if <math>(x,y)</math> is on the angle bisector, then <math>x=\frac{y}{m_1}</math>. Now let the centers of the two relevant circles be <math>(a/m_1 , a)</math> and <math>(b/m_1 , b)</math> for some positive reals <math>a</math> and <math>b</math>. These two circles are tangent to the <math>x</math>-axis, so the radii of the circles are <math>a</math> and <math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
 
Let the smaller angle between the <math>x</math>-axis and the line <math>y=mx</math> be <math>\theta</math>. Note that the centers of the two circles lie on the angle bisector of the angle between the <math>x</math>-axis and the line <math>y=mx</math>. Also note that if <math>(x,y)</math> is on said angle bisector, we have that <math>\frac{y}{x}=\tan{\frac{\theta}{2}}</math>.  Let <math>\tan{\frac{\theta}{2}}=m_1</math>, for convenience. Therefore if <math>(x,y)</math> is on the angle bisector, then <math>x=\frac{y}{m_1}</math>. Now let the centers of the two relevant circles be <math>(a/m_1 , a)</math> and <math>(b/m_1 , b)</math> for some positive reals <math>a</math> and <math>b</math>. These two circles are tangent to the <math>x</math>-axis, so the radii of the circles are <math>a</math> and <math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
  
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Expanding these and manipulating terms gives
 
Expanding these and manipulating terms gives
  
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<cmath>\frac{1}{m_1}^2a^2-[(18/m_1)+12]a+117=0</cmath>
+
<cmath>\frac{1}{m_1^2}a^2-[(18/m_1)+12]a+117=0</cmath>
  
−
<cmath>\frac{1}{m_1}^2b^2-[(18/m_1)+12]b+117=0</cmath>
+
<cmath>\frac{1}{m_1^2}b^2-[(18/m_1)+12]b+117=0</cmath>
  
 
It follows that <math>a</math> and <math>b</math> are the roots of the quadratic
 
It follows that <math>a</math> and <math>b</math> are the roots of the quadratic
  
−
<cmath>\frac{1}{m_1}^2x^2-[(18/m_1)+12]x+117=0</cmath>
+
<cmath>\frac{1}{m_1^2}x^2-[(18/m_1)+12]x+117=0</cmath>
  
 
It follows from Vieta's Formulas that the product of the roots of this quadratic is <math>117m_1^2</math>, but we were also given that the product of the radii was 68. Therefore <math>68=117m_1^2</math>, or <math>m_1^2=\frac{68}{117}</math>. Note that the half-angle formula for tangents is
 
It follows from Vieta's Formulas that the product of the roots of this quadratic is <math>117m_1^2</math>, but we were also given that the product of the radii was 68. Therefore <math>68=117m_1^2</math>, or <math>m_1^2=\frac{68}{117}</math>. Note that the half-angle formula for tangents is
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It then follows that <math>m=\tan{\theta}=\frac{12\sqrt{221}}{49}</math>. Therefore <math>a=12</math>, <math>b=221</math>, and <math>c=49</math>. The desired answer is then <math>12+221+49=\boxed{282}</math>.
 
It then follows that <math>m=\tan{\theta}=\frac{12\sqrt{221}}{49}</math>. Therefore <math>a=12</math>, <math>b=221</math>, and <math>c=49</math>. The desired answer is then <math>12+221+49=\boxed{282}</math>.
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 +
== Solution 2 (Alcumus)==
 +
Let <math>r_1</math> and <math>r_2</math> be the radii of the circles. Then the centers of the circles are of the form <math>(kr_1,r_1)</math> and <math>(kr_2,r_2)</math> for the same constant <math>k,</math> since the two centers are collinear with the origin. Since <math>(9,6)</math> lies on both circles,
 +
<cmath>(kr_i - 9)^2 + (r_i - 6)^2 = r^2,</cmath>where <math>r_i</math> represents either radius. Expanding, we get
 +
<cmath>k^2 r^2 - (18k + 12) r + 117 = 0.</cmath>We are told the product of the circles is 68, so by Vieta's formulas, <math>\frac{117}{k^2} = 68.</math> Hence, <math>k^2 = \frac{117}{68},</math> and <math>k = \sqrt{\frac{117}{68}}.</math>
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 +
<asy>
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unitsize(0.25 cm);
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pair[] O;
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real[] r;
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pair P;
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r[1] = 4.096;
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r[2] = 16.6;
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O[1] = (r[1]/(2/3*sqrt(17/13)),r[1]);
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O[2] = (r[2]/(2/3*sqrt(17/13)),r[2]);
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P = reflect(O[1],O[2])*(9,6);
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draw(Circle(O[1],r[1]));
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//draw(Circle(O[2],r[2]));
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draw(arc(O[2],r[2],130,300));
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draw((0,0)--(8,12*sqrt(221)/49*8));
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draw((0,0)--(30,0));
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draw((0,0)--O[1]--(O[1].x,0));
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draw(O[1]--(O[1] + reflect((0,0),(10,12*sqrt(221)/49*10))*(O[1]))/2);
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label("$y = mx$", (8,12*sqrt(221)/49*8), N);
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dot("$(9,6)$", (9,6), NE);
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dot("$(kr,r)$", O[1], N);
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dot(P,red);
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</asy>
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 +
Since the circle is tangent to the line <math>y = mx,</math> the distance from the center <math>(kr,r)</math> to the line is <math>r.</math> We can write <math>y = mx</math> as <math>y - mx = 0,</math> so from the distance formula,
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<cmath>\frac{|r - krm|}{\sqrt{1 + m^2}} = r.</cmath>Squaring both sides, we get
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<cmath>\frac{(r - krm)^2}{1 + m^2} = r^2,</cmath>so <math>(r - krm)^2 = r^2 (1 + m^2).</math> Since <math>r \neq 0,</math> we can divide both sides by r, to get
 +
<cmath>(1 - km)^2 = 1 + m^2.</cmath>Then <math>1 - 2km + k^2 m^2 = 1 + m^2,</math> so <math>m^2 (1 - k^2) + 2km = 0.</math> Since <math>m \neq 0,</math>
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<cmath>m(1 - k^2) + 2k = 0.</cmath>Hence,
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<cmath>m = \frac{2k}{k^2 - 1} = \frac{2 \sqrt{\frac{117}{68}}}{\frac{117}{68} - 1} = \frac{12 \sqrt{221}}{49}.</cmath>
 +
Our answer is thus <math>12 + 221 + 49 = \boxed{282}</math>
 +
 +
== Solution 3 ==
 +
Let the centers of <math>C_1</math> and <math>C_2</math> be <math>A</math> and <math>B</math>, respectively, and let the point <math>(9, 6)</math> be <math>P</math>.
 +
 +
Because both <math>C_1</math> and <math>C_2</math> are tangent to the x-axis, and both of them pass through <math>P</math>, both <math>A</math> and <math>B</math> must be equidistant from <math>P</math> and the x-axis. Therefore, they must both be on the parabola with <math>P</math> as the focus and the x-axis as the directrix. Since the coordinates of <math>P</math> is <math>(9, 6)</math>, we see that this parabola is the graph of the function <cmath>y=\frac{1}{12}(x-9)^2+3=\frac{1}{12}x^2-\frac{3}{2}x+\frac{39}{4}.</cmath>
 +
 +
Let <math>AB</math> be <math>y=kx</math>. Because <math>C_1</math> and <math>C_2</math> are both tangent to the x-axis, the y-coordinates of <math>A</math> and <math>B</math> are <math>r_1</math> and <math>r_2</math>, respectively, so the x-coordinates of <math>A</math> and <math>B</math> are <math>\frac{r_1}{k}</math> and <math>\frac{r_2}{k}</math>. But since <math>A</math> and <math>B</math> are also on the graph of the function <math>y=\frac{1}{12}x^2-\frac{3}{2}x+\frac{39}{4}</math>, the x-coordinates of <math>A</math> and <math>B</math> are also the roots of the equation <math>kx=\frac{1}{12}x^2-\frac{3}{2}x+\frac{39}{4}</math>, and by Vieta's Formulas, their product is <math>\frac{\frac{39}{4}}{\frac{1}{12}}=117</math>. So we have <math>\frac{r_1}{k}\cdot \frac{r_2}{k}=117</math>.
 +
 +
We are also given that <math>r_1r_2=68</math>, so <math>k^2=\frac{r_1r_2}{117}=\frac{68}{117}</math>, which means that <math>k=\sqrt{\frac{68}{117}}</math>. Note that the line <math>AB</math> is the angle bisector of the angle between the line <math>y=mx</math> and the x-axis. Therefore, we apply the double-angle formula for tangents and get <cmath>m=\frac{2k}{1-k^2}=\frac{2\sqrt{\frac{68}{117}}}{1-\frac{68}{117}}=\frac{12\sqrt{221}}{49}.</cmath> Thus, the answer is <math>12+221+49=\boxed{282}</math>.
 +
 +
== Sidenote ==
 +
The two circles are centered at <cmath>\left (\frac{1}{13}\left (117 + 4 \sqrt{221} \pm 2 \sqrt{13(18 \sqrt{221} - 49)} \right), \frac{2}{39} \left(68 + 9 \sqrt {221} \pm 2 \sqrt{17(18 \sqrt{221} - 49)} \right) \right)</cmath>
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2002|n=II|num-b=14|after=Last Question}}
 
{{AIME box|year=2002|n=II|num-b=14|after=Last Question}}
 +
 +
[[Category: Intermediate Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 16:46, 23 June 2024

Problem

Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$, and the product of the radii is $68$. The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt {b}/c$, where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a + b + c$.

Solution 1

Let the smaller angle between the $x$-axis and the line $y=mx$ be $\theta$. Note that the centers of the two circles lie on the angle bisector of the angle between the $x$-axis and the line $y=mx$. Also note that if $(x,y)$ is on said angle bisector, we have that $\frac{y}{x}=\tan{\frac{\theta}{2}}$. Let $\tan{\frac{\theta}{2}}=m_1$, for convenience. Therefore if $(x,y)$ is on the angle bisector, then $x=\frac{y}{m_1}$. Now let the centers of the two relevant circles be $(a/m_1 , a)$ and $(b/m_1 , b)$ for some positive reals $a$ and $b$. These two circles are tangent to the $x$-axis, so the radii of the circles are $a$ and $b$ respectively. We know that the point $(9,6)$ is a point on both circles, so we have that

\[(9-\frac{a}{m_1})^2+(6-a)^2=a^2\]

\[(9-\frac{b}{m_1})^2+(6-b)^2=b^2\]

Expanding these and manipulating terms gives

\[\frac{1}{m_1^2}a^2-[(18/m_1)+12]a+117=0\]

\[\frac{1}{m_1^2}b^2-[(18/m_1)+12]b+117=0\]

It follows that $a$ and $b$ are the roots of the quadratic

\[\frac{1}{m_1^2}x^2-[(18/m_1)+12]x+117=0\]

It follows from Vieta's Formulas that the product of the roots of this quadratic is $117m_1^2$, but we were also given that the product of the radii was 68. Therefore $68=117m_1^2$, or $m_1^2=\frac{68}{117}$. Note that the half-angle formula for tangents is

\[\tan{\frac{\theta}{2}}=\sqrt{\frac{1-\cos{\theta}}{1+\cos{\theta}}}\]

Therefore

\[\frac{68}{117}=\frac{1-\cos{\theta}}{1+\cos{\theta}}\]

Solving for $\cos{\theta}$ gives that $\cos{\theta}=\frac{49}{185}$. It then follows that $\sin{\theta}=\sqrt{1-\cos^2{\theta}}=\frac{12\sqrt{221}}{185}$.

It then follows that $m=\tan{\theta}=\frac{12\sqrt{221}}{49}$. Therefore $a=12$, $b=221$, and $c=49$. The desired answer is then $12+221+49=\boxed{282}$.

Solution 2 (Alcumus)

Let $r_1$ and $r_2$ be the radii of the circles. Then the centers of the circles are of the form $(kr_1,r_1)$ and $(kr_2,r_2)$ for the same constant $k,$ since the two centers are collinear with the origin. Since $(9,6)$ lies on both circles, \[(kr_i - 9)^2 + (r_i - 6)^2 = r^2,\]where $r_i$ represents either radius. Expanding, we get \[k^2 r^2 - (18k + 12) r + 117 = 0.\]We are told the product of the circles is 68, so by Vieta's formulas, $\frac{117}{k^2} = 68.$ Hence, $k^2 = \frac{117}{68},$ and $k = \sqrt{\frac{117}{68}}.$

[asy] unitsize(0.25 cm); pair[] O; real[] r; pair P; r[1] = 4.096; r[2] = 16.6; O[1] = (r[1]/(2/3*sqrt(17/13)),r[1]); O[2] = (r[2]/(2/3*sqrt(17/13)),r[2]); P = reflect(O[1],O[2])*(9,6); draw(Circle(O[1],r[1])); //draw(Circle(O[2],r[2])); draw(arc(O[2],r[2],130,300)); draw((0,0)--(8,12*sqrt(221)/49*8)); draw((0,0)--(30,0)); draw((0,0)--O[1]--(O[1].x,0)); draw(O[1]--(O[1] + reflect((0,0),(10,12*sqrt(221)/49*10))*(O[1]))/2); label("$y = mx$", (8,12*sqrt(221)/49*8), N); dot("$(9,6)$", (9,6), NE); dot("$(kr,r)$", O[1], N); dot(P,red); [/asy]

Since the circle is tangent to the line $y = mx,$ the distance from the center $(kr,r)$ to the line is $r.$ We can write $y = mx$ as $y - mx = 0,$ so from the distance formula, \[\frac{|r - krm|}{\sqrt{1 + m^2}} = r.\]Squaring both sides, we get \[\frac{(r - krm)^2}{1 + m^2} = r^2,\]so $(r - krm)^2 = r^2 (1 + m^2).$ Since $r \neq 0,$ we can divide both sides by r, to get \[(1 - km)^2 = 1 + m^2.\]Then $1 - 2km + k^2 m^2 = 1 + m^2,$ so $m^2 (1 - k^2) + 2km = 0.$ Since $m \neq 0,$ \[m(1 - k^2) + 2k = 0.\]Hence, \[m = \frac{2k}{k^2 - 1} = \frac{2 \sqrt{\frac{117}{68}}}{\frac{117}{68} - 1} = \frac{12 \sqrt{221}}{49}.\] Our answer is thus $12 + 221 + 49 = \boxed{282}$

Solution 3

Let the centers of $C_1$ and $C_2$ be $A$ and $B$, respectively, and let the point $(9, 6)$ be $P$.

Because both $C_1$ and $C_2$ are tangent to the x-axis, and both of them pass through $P$, both $A$ and $B$ must be equidistant from $P$ and the x-axis. Therefore, they must both be on the parabola with $P$ as the focus and the x-axis as the directrix. Since the coordinates of $P$ is $(9, 6)$, we see that this parabola is the graph of the function \[y=\frac{1}{12}(x-9)^2+3=\frac{1}{12}x^2-\frac{3}{2}x+\frac{39}{4}.\]

Let $AB$ be $y=kx$. Because $C_1$ and $C_2$ are both tangent to the x-axis, the y-coordinates of $A$ and $B$ are $r_1$ and $r_2$, respectively, so the x-coordinates of $A$ and $B$ are $\frac{r_1}{k}$ and $\frac{r_2}{k}$. But since $A$ and $B$ are also on the graph of the function $y=\frac{1}{12}x^2-\frac{3}{2}x+\frac{39}{4}$, the x-coordinates of $A$ and $B$ are also the roots of the equation $kx=\frac{1}{12}x^2-\frac{3}{2}x+\frac{39}{4}$, and by Vieta's Formulas, their product is $\frac{\frac{39}{4}}{\frac{1}{12}}=117$. So we have $\frac{r_1}{k}\cdot \frac{r_2}{k}=117$.

We are also given that $r_1r_2=68$, so $k^2=\frac{r_1r_2}{117}=\frac{68}{117}$, which means that $k=\sqrt{\frac{68}{117}}$. Note that the line $AB$ is the angle bisector of the angle between the line $y=mx$ and the x-axis. Therefore, we apply the double-angle formula for tangents and get \[m=\frac{2k}{1-k^2}=\frac{2\sqrt{\frac{68}{117}}}{1-\frac{68}{117}}=\frac{12\sqrt{221}}{49}.\] Thus, the answer is $12+221+49=\boxed{282}$.

Sidenote

The two circles are centered at \[\left (\frac{1}{13}\left (117 + 4 \sqrt{221} \pm 2 \sqrt{13(18 \sqrt{221} - 49)} \right), \frac{2}{39} \left(68 + 9 \sqrt {221} \pm 2 \sqrt{17(18 \sqrt{221} - 49)} \right) \right)\]

See also

2002 AIME II (Problems • Answer Key • Resources)
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