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Difference between revisions of "2015 AIME II Problems/Problem 15"

(→‎Problem)
(Fixed an incorrect equation in Solution 2 and added link to Homothety)
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==Solution 2==
 
==Solution 2==
−
By homothety, we deduce that <math>AE = 4 AD</math>. (The proof can also be executed by similar triangles formed from dropping perpendiculars from the centers of <math>P</math> and <math>Q</math> to <math>l</math>.) Therefore, our equality of area condition, or the equality of base times height condition, reduces to the fact that the distance from <math>B</math> to <math>l</math> is four times that from <math>C</math> to <math>l</math>. Let the distance from <math>C</math> be <math>x</math> and the distance from <math>B</math> be <math>4x</math>.
+
By [[homothety]], we deduce that <math>AE = 4 AD</math>. (The proof can also be executed by similar triangles formed from dropping perpendiculars from the centers of <math>P</math> and <math>Q</math> to <math>l</math>.) Therefore, our equality of area condition, or the equality of base times height condition, reduces to the fact that the distance from <math>B</math> to <math>l</math> is four times that from <math>C</math> to <math>l</math>. Let the distance from <math>C</math> be <math>x</math> and the distance from <math>B</math> be <math>4x</math>.
  
 
Let <math>P</math> and <math>Q</math> be the centers of their respective circles. Then dropping a perpendicular from <math>P</math> to <math>Q</math> creates a <math>3-4-5</math> right triangle, from which <math>BC = 4</math> and, if <math>\alpha = \angle{AQC}</math>, that <math>\cos \alpha = \dfrac{3}{5}</math>. Then <math>\angle{BPA} = 180^\circ - \alpha</math>, and the Law of Cosines on triangles <math>APB</math> and <math>AQC</math> gives <math>AB = \dfrac{4}{\sqrt{5}}</math> and <math>AC = \dfrac{8}{\sqrt{5}}.</math>
 
Let <math>P</math> and <math>Q</math> be the centers of their respective circles. Then dropping a perpendicular from <math>P</math> to <math>Q</math> creates a <math>3-4-5</math> right triangle, from which <math>BC = 4</math> and, if <math>\alpha = \angle{AQC}</math>, that <math>\cos \alpha = \dfrac{3}{5}</math>. Then <math>\angle{BPA} = 180^\circ - \alpha</math>, and the Law of Cosines on triangles <math>APB</math> and <math>AQC</math> gives <math>AB = \dfrac{4}{\sqrt{5}}</math> and <math>AC = \dfrac{8}{\sqrt{5}}.</math>
  
 
Now, using the Pythagorean Theorem to express the length of the projection of <math>BC</math> onto line <math>l</math> gives
 
Now, using the Pythagorean Theorem to express the length of the projection of <math>BC</math> onto line <math>l</math> gives
−
<cmath>\sqrt{\frac{16}{5} - x^2} + \sqrt{\frac{64}{5} - x^2} = \sqrt{16 - 9x^2}.</cmath>
+
<cmath>\sqrt{\frac{16}{5} - 16x^2} + \sqrt{\frac{64}{5} - x^2} = \sqrt{16 - 9x^2}.</cmath>
 
Squaring and simplifying gives
 
Squaring and simplifying gives
 
<cmath>\sqrt{\left(\frac{1}{5} - x^2\right)\left(\frac{64}{5} - x^2\right)} = x^2,</cmath>
 
<cmath>\sqrt{\left(\frac{1}{5} - x^2\right)\left(\frac{64}{5} - x^2\right)} = x^2,</cmath>

Revision as of 15:24, 17 September 2015

Problem

Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[asy] import cse5; pathpen=black; pointpen=black; size(6cm); pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689); filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7)); filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7)); D(CR((0,1),1)); D(CR((4,4),4,150,390)); D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5)); D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0)); D(MP("E",E,N)); [/asy]

Hint

This is a #15 on an AIME, so it must be difficult. Indeed, there are two possible approaches (both of them very computational): coordinate geometry, or regular Euclidean geometry combined with a bit of trigonometry.

Solution 1

[asy] unitsize(35); draw(Circle((-1,0),1)); draw(Circle((4,0),4)); pair A,O_1, O_2, B,C,D,E,N,K,L,X,Y; A=(0,0);O_1=(-1,0);O_2=(4,0);B=(-24/15,-12/15);D=(-8/13,12/13);E=(32/13,-48/13);C=(24/15,-48/15);N=extension(E,B,O_2,O_1);K=foot(B,O_1,N);L=foot(C,O_2,N);X=foot(B,A,D);Y=foot(C,E,A); label("$A$",A,NE);label("$O_1$",O_1,NE);label("$O_2$",O_2,NE);label("$B$",B,SW);label("$C$",C,SW);label("$D$",D,NE);label("$E$",E,NE);label("$N$",N,W);label("$K$",(-24/15,0.2));label("$L$",(24/15,0.2));label("$n$",(-0.8,-0.12));label("$p$",((29/15,-48/15)));label("$\mathcal{P}$",(-1.6,1.1));label("$\mathcal{Q}$",(6,4)); draw(A--B--D--cycle);draw(A--E--C--cycle);draw(C--N);draw(O_2--N);draw(O_1--B,dashed);draw(O_2--C,dashed); dot(O_1);dot(O_2); draw(rightanglemark(O_1,B,N,5));draw(rightanglemark(O_2,C,N,5));draw(C--L,dashed);draw(B--K,dashed);draw(C--Y);draw(B--X); [/asy]

Call $O_1$ and $O_2$ the centers of circles $\mathcal{P}$ and $\mathcal{Q}$, respectively, and call $K$ and $L$ the feet of the altitudes from $B$ to $O_1N$ and $C$ to $O_2N$, respectively. Extend $CB$ and $O_2O_1$ to meet at point $N$. Using the fact that $\triangle{O_1BN} \sim \triangle{O_2CN}$ and setting $NO_1 = k$, we have that $\frac{k+5}{k} = \frac{4}{1} \implies k=\frac{5}{3}$. We can do some more length chasing using triangles similar to $OBN$ to get that $AK = AL = \frac{24}{15}$, $BK = \frac{12}{15}$, and $CL = \frac{48}{15}$. Now, consider the circles $\mathcal{P}$ and $\mathcal{Q}$ on the coordinate plane, where $A$ is the origin. If the line $\ell$ through $A$ intersects $\mathcal{P}$ at $D$ and $\mathcal{Q}$ at $E$, then $4 \cdot DA = AE$. To verify this, notice that $\triangle{AO_1D} \sim \triangle{EO_2A}$ from the fact that both triangles are isosceles with $\angle{O_1AD} \cong \angle{O_2AE}$, which are corresponding angles. Since $O_2A = 4\cdot O_1A$, we can conclude that $4 \cdot DA = AE$.


Hence, we need to find the slope $m$ of line $\ell$ such that the perpendicular distance $n$ from $B$ to $AD$ is four times the perpendicular distance $p$ from $C$ to $AE$. This will mean that the product of the bases and heights of triangles $ACE$ and $DBA$ will be equal, which in turn means that their areas will be equal. Let the line $\ell$ have the equation $y = -mx \implies mx + y = 0$, and let $m$ be a positive real number so that the negative slope of $\ell$ is preserved. Then, the coordinates of $B$ are $(x_B, y_B) = \left(\frac{-24}{15}, \frac{-12}{15}\right)$, and the coordinates of $C$ are $(x_C, y_C) = \left(\frac{24}{15}, \frac{-48}{15}\right)$. Using the point-to-line distance formula and the fact that $n = 4p$, we have \[\frac{|mx_B + 1(y_B) + 0|}{\sqrt{m^2 + 1}} = \frac{4|mx_C + 1(y_C) + 0|}{\sqrt{m^2 + 1}}\] \[\implies |mx_B + y_B| = 4|mx_C + y_C| \implies \left|\frac{-24m}{15} + \frac{-12}{15}\right| = 4\left|\frac{24m}{15} + \frac{-48}{15}\right|.\] Since $m$ takes on a positive value, we must switch the signs of all terms in this equation when we get rid of the absolute value signs. We then have \[\frac{6m}{15} + \frac{3}{15} = \frac{48}{15} - \frac{24m}{15} \implies 2m = 3 \implies m = \frac{3}{2}.\] Thus, the equation of $\ell$ is $y = -\frac{3}{2}x$.


Then we can find the coordinates of $D$ by finding the point $(x,y)$ other than $A = (0,0)$ where the circle $\mathcal{P}$ intersects $\ell$. $\mathcal{P}$ can be represented with the equation $(x + 1)^2 + y^2 = 1$, and substituting $y = -\frac{3}{2}x$ into this equation yields \[x^2 + 2x + 1 + \frac{9}{4}x^2 = 1 \implies \frac{13}{4}x^2 + 2x = 0 \implies x = 0, -\frac{8}{13}.\] Discarding $x = 0$, the $y$-coordinate of $D$ is $-\frac{3}{2} \cdot -\frac{8}{13} = \frac{12}{13}$. The distance from $D$ to $A$ is then $\sqrt{\left(\frac{-8}{13}\right)^2 + \left(\frac{12}{13}\right)^2} = \frac{4}{\sqrt{13}}.$ The perpendicular distance from $B$ to $AD$ or the height of $\triangle{DBA}$ is $\frac{|\frac{3}{2}\cdot\frac{-24}{15} + \frac{-12}{15} + 0|}{\sqrt{\frac{3}{2}^2 + 1}} = \frac{\frac{48}{15}}{\frac{\sqrt{13}}{2}} = \frac{32}{5\sqrt{13}}.$ Finally, the common area is $\frac{1}{2}\left(\frac{32}{5\sqrt{13}} \cdot \frac{4}{\sqrt{13}}\right) = \frac{64}{65}$, and $m + n = 64 + 65 = \boxed{129}$.

Solution 2

By homothety, we deduce that $AE = 4 AD$. (The proof can also be executed by similar triangles formed from dropping perpendiculars from the centers of $P$ and $Q$ to $l$.) Therefore, our equality of area condition, or the equality of base times height condition, reduces to the fact that the distance from $B$ to $l$ is four times that from $C$ to $l$. Let the distance from $C$ be $x$ and the distance from $B$ be $4x$.

Let $P$ and $Q$ be the centers of their respective circles. Then dropping a perpendicular from $P$ to $Q$ creates a $3-4-5$ right triangle, from which $BC = 4$ and, if $\alpha = \angle{AQC}$, that $\cos \alpha = \dfrac{3}{5}$. Then $\angle{BPA} = 180^\circ - \alpha$, and the Law of Cosines on triangles $APB$ and $AQC$ gives $AB = \dfrac{4}{\sqrt{5}}$ and $AC = \dfrac{8}{\sqrt{5}}.$

Now, using the Pythagorean Theorem to express the length of the projection of $BC$ onto line $l$ gives \[\sqrt{\frac{16}{5} - 16x^2} + \sqrt{\frac{64}{5} - x^2} = \sqrt{16 - 9x^2}.\] Squaring and simplifying gives \[\sqrt{\left(\frac{1}{5} - x^2\right)\left(\frac{64}{5} - x^2\right)} = x^2,\] and squaring and solving gives $x = \dfrac{8}{5\sqrt{13}}.$

By the Law of Sines on triangle $ABD$, we have \[\frac{BD}{\sin A} = 2.\] But we know $\sin A = \dfrac{4x}{AB}$, and so a small computation gives $BD = \dfrac{16}{\sqrt{65}}.$ The Pythagorean Theorem now gives \[AD = \sqrt{BD^2 - (4x)^2} + \sqrt{AB^2 - (4x)^2} = \frac{4}{\sqrt{13}},\] and so the common area is $\dfrac{1}{2} \cdot \frac{4}{\sqrt{13}} \cdot \frac{32}{5\sqrt{13}} = \frac{64}{65}.$ The answer is $\boxed{129}.$

See also

2015 AIME II (Problems • Answer Key • Resources)
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