MathDB

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]

Problem Statement