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July 20th, 2017, 08:59 PM   #1
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Need help to solve this Geometry question !!!

Hello Everyone,

I need to find the value of x.

Attached Images
 1.JPG (78.8 KB, 25 views)

 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
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Quote:
 Originally Posted by skipjack As a function of r1, r2 and a?
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
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Quote:
 Originally Posted by skipjack 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.
Yeah, if it can be solved using co-ordinates then please do it, actually I can't figure it out how to solve this problem using co-ordinates.

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).
Attached Images
 1.jpg (63.8 KB, 15 views)

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
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Quote:
 Originally Posted by skipjack 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?
Hi skipjack,

From your equation I'm getting value of
x° = 45° - (sin-¹ ( (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

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. Thanks from sagar233
July 23rd, 2017, 06:04 AM   #9
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Quote:
 Originally Posted by skipjack 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.
Hi skipjack,

Thanks a lot for this solution.

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