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Q.

Circles of radii 36 and 9 touch externally. The radius of the circle which touches the two circles externally and also their common tangent is :


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a

2

b

4

c

17

d

18 

answer is B.

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Detailed Solution

Question ImageLet the center of the small circle with radius 9 as A and center of the large circle radius 36 as C. B and D are the points of contact with their common tangent.
AB=9,CD=36.
Let the center of the circle which touches both the small and large circle as O and the point of contact of this circle on the tangent is G. So its radius is OG(r).
We produce the line segment MN which passes through O and parallel and equal to BD. We also produce the line segment AL which is parallel and equal to BD. We know that the radius of a circle is perpendicular to its tangent. So BD as well as its parallel lines MN and AL will be perpendicular to both radii AB and CD.
Let BG = x, BD = d
⇒ GD = d−x
⇒ BD = AL = MN = d, OM = d−x, ON = x.
AC = AB+CD = 36+9 = 45
CL = CD−LD = CD−AB = 36−9 = 27.
In the right-angled triangle ALC,
AC2=AL2+LC2
⇒452=d2-272
⇒d=36
In the right-angled triangle AON, AO = 9 + OG = 9 + r and AN = AB − NB = AB − OG = 9−r.
AO2=AN2+NO2
⇒(9+r)2=(9-r)2+x2
⇒x=6r
In the right-angled triangle COM, CO=CD+OG=36+r and CM=CD−DM=CD−OG=36−r.
CO2=CM2+MO2
⇒(36+r)2=(36-r)2+(d-x)2
⇒d-x=12r
⇒36-6r=12r
⇒r=2
⇒r=4
So, Option 2 is correct.
 
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