Two spherical balls lie on the ground touching. If one of

Question

Two spherical balls lie on the ground touching. If one of the balls has a radius of 8 cm, and the point of contact is 10 cm above the ground, what is the radius of the other ball?

(A) 18 cm 
(B) 40/3 cm 
(C) 25/2 cm 
(D) 13 cm 
(E) none of the these

jamifahad · Answer

See the attached image. smaller circle has radius 8. Consider two triangle. One with hypotenuse 8 and another with hypotenuse 8+x

In smaller triangle, sides are  8,2,2\sqrt{15} 
sin@ = 2/8=1/4

In larger triangle, sides are  8+x, 2\sqrt{15}+10, x-8 
sin@= x-8/x+8
1/4=x-8/x+8
x=40/3.

amit2k9 · Answer

similar triangle properties..

2/r+8 = 8/r-8

giving r = 40/3.

maheshsrini · Answer

How do we know the line joining the two centre's passes through the point of intersection of the two circles?.

Can somebody please clarify this?.

KarishmaB · Answer

For the moment, just think of the two circles touching each other at point, say P, as shown in the diagram in the post by jamifahad.

Now, say, you draw a tangent at P, the point of intersection. This will be a common tangent to both the circles at the point P. Join the center of each circle with P. We know that the line joining the center to the point of tangency will be perpendicular to the tangent. 
Hence they will form an angle of 180 degrees and will therefore be straight.

sudhir18n · Answer

I think using similar triangle property is the best 
From the diagram below , we know 
triangle CDA is similar triangle BEA
therefore 
BE/EA= CD/DA..... (DA is the radius we are looking for)
2/8 = (r-8)/(r+8)
solve to get r = 40/3

maheshsrini · Answer

yeah, got it. Thanks Karishma.

aditya16 · Answer

I have a query.
In the ques it it stated that "the point of contact is 10 cm above the ground". But in the images of solutions proposed, the value 10 is suggested as distance between point of contacts of circles with the ground. 
Can someone clarify what I am missing in comprehending the problem?

nayas96 · Answer

KarishmaB  Kindly please help

KarishmaB · Answer

We know that the balls are lying on the ground and touching each other. The point of contact is the point where the two balls meet. They will touch in the air, not on the ground because they are spherical. This point of contact of the two balls is 10 cm above the ground.

nayas96

nayas96 · Answer

KarishmaB  Thank you for your reply. But in the above figure, the similarity ratio is based on the fact that the horizontal distance between the 2 radius is 10 cm.

Kindly please clarify

KarishmaB · Answer

The green line is 10 cm. Screenshot 2022-04-24 at 12.01.02.png Using similar triangles for the right triangles formed by joining the centres of the circles, 8/(8+x) = 2/(x - 8) 8x - 64 = 16 + 2x x = 40/3 Answer (B)

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Two spherical balls lie on the ground touching. If one of

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Two spherical balls lie on the ground touching. If one of

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