## Saturday, May 9, 2009

### Problem 289: Tangent circles, Radius, Perpendicular, Tangent

Proposed Problem

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See also:
Problem 289: Tangent circles, Radius, Perpendicular, Tangent
Collection of Geometry Problems

Level: High School, SAT Prep, College geometry

#### 3 comments:

1. Let A be (0,0) and AD the x-axis, the radius of circle B = r. Hence B:(((a^2+2ar)^(1/2),r) and E:(0,a). Let F be (x1,y1). Thus we have
x1-(a^2+2ra)^(1/2))^2+(y1-r)^2=r^2 since F lies on circle B. Considering slopes of EF & BF we get:-x1/(y1-a)=(y1-r)/(x1-(a^2+2ra)^(1/2)) and
(EF)^2 = (y1-a)^2 + x1^2. On solving the equations obtained we get: EF=V2*a or x=V2*a
Ajit: ajitathle@gmail.com

2. From triangle BFE we can say that: x^2=BE^2-r^2 if r is the radius of the circle B. But B is ((a^2+2ar)^(1/2),r)as may easily ascertained and A is (0,a). Hence we've: x^2 = [(a^2+2ar)^(1/2)-0]^2+(r-a)^2 - r^2 using the distance formula. This easily gives x^2 = 2a^2 w/o having to worry about the co-ords. of F.
Ajit

3. Let radius of circle B be b.
AB=a+b
BD=b
AD^2 = (a+b)^2 - b^2 = a*a + 2ab
Now draw a line parallel to AD from B to AE which touches AE at say M. Now ME = a-b
EB^2 = (a-b)^2 + a^2 +2ab = 2*a*a + b*b
Now EF^2 + b^2 = EB^2
EF^2 = 2a*a + b*b -b*b = 2a*a
EF = a*sqrt(2).