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July 20th, 2017, 08:59 PM  #1 
Senior Member Joined: Jun 2017 From: India Posts: 234 Thanks: 6  Need help to solve this Geometry question !!!
Hello Everyone, Please help me solving a geometry related problem which I've attached. I need to find the value of x. Thanks in Advance. 
July 20th, 2017, 09:36 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,168 Thanks: 1640 
As a function of r1, r2 and a?

July 20th, 2017, 10:22 PM  #3 
Senior Member Joined: Jun 2017 From: India Posts: 234 Thanks: 6  Yes, r1, r2 and a are any numbers greater than 0. I need to find the value of x in terms of r1, r2 and a. Last edited by skipjack; July 21st, 2017 at 05:27 AM. 
July 21st, 2017, 05:25 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,168 Thanks: 1640 
Can you use coordinate geometry? Start by supplying a label (O, A, B, etc.) for each point where lines (or curves) meet in the diagram. It's then fairly easy to obtain an equation that x satisfies, etc.

July 21st, 2017, 07:59 PM  #5  
Senior Member Joined: Jun 2017 From: India Posts: 234 Thanks: 6  Quote:
But solving it using geometry/trigo/algebra will be more useful. Actually I tried using geometry/trigonometry/algebra but got stuck after finding the value of AB (see attached image). Last edited by sagar233; July 21st, 2017 at 08:08 PM.  
July 22nd, 2017, 03:41 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,168 Thanks: 1640 
AB has equation x  y  a = 0 (the x here isn't the angle in the diagram). OP = r1  r2, so P has coordinates ((r1  r2)cos(x°), (r1  r2)sin(x°)). Hence r2 = (r1  r2)cos(x°)  (r1  r2)sin(x°)  a/√2 = (r1  r2)sin(45°  x°)  a/√2. Can you proceed from there? 
July 23rd, 2017, 03:27 AM  #7  
Senior Member Joined: Jun 2017 From: India Posts: 234 Thanks: 6  Quote:
Thanks for reply. From your equation I'm getting value of x° = 45°  (sin¹ ( (r2+a/√2) / (r1  r2) ) ) But this equation is not giving correct answer. For Example: if r1 = 10, r2 = 2 and a = 5, then answer x° should be ~33.934 but by above equation I'm getting ~1.216 please check it. Last edited by sagar233; July 23rd, 2017 at 03:29 AM.  
July 23rd, 2017, 05:18 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,168 Thanks: 1640 
The expression (r1  r2)sin(45°  x°)  a/√2 is negative for your values of a, r1, r2 and x $\small\approx$ 33.934, so use a/√2  (r1  r2)sin(45°  x°) for its absolute value. Your original calculated value of x corresponds to a diagram in which the smaller circle lies on the other side of the line AB. 
July 23rd, 2017, 06:04 AM  #9  
Senior Member Joined: Jun 2017 From: India Posts: 234 Thanks: 6  Quote:
Thanks a lot for this solution.  

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geometry, question, solve, trigonometry 
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