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Two perpendicular diameters AB and CD of a circle intersect at the center O. A second circle is tangent to the first circle at B, and intersects AO and CO at P and Q, respectively. If AP = 10 and CQ = 7, then find the radius of the larger circle.

 Dec 14, 2019
 #1
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The radius of the larger circle is 16/3.

 Dec 14, 2019
 #2
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See the following image :

 

 

Let the radius of the large circle  = R

 

Due to  symmetry, QF = EF   = R - 7

 

And PF  = R - 10

And FB  =  R

 

And by the intersecting chord theorem in the smaller circle

 

(QF)(EF)  = (PF)(FB)

 

(R - 7) (R - 7)  =(R- 10)(R)

 

(R - 7) (R -7) = R^2 - 10R

 

R^2 - 14R + 49 = R^2 - 10r     rearrange  as

 

49  = 4R

 

R =  49/4  =  12.25 =  the  radius of the larger circle

 

 

cool cool cool

 Dec 15, 2019

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