AIME 2002 II · 第 15 题

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 $k = \sqrt{\frac{117}{68}}.$

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

$$\left (\frac{1}{13}\left (117 + 4 \sqrt{221} \pm 2 \sqrt{13(18 \sqrt{221} - 49)} \right), \frac{2}{39} \left(68 + 9 \sqrt {221} \pm 2 \sqrt{17(18 \sqrt{221} - 49)} \right) \right)$$