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November 19th, 2013, 10:49 AM  #1 
Member Joined: Nov 2013 Posts: 34 Thanks: 1  Calculate the center of a circle given a tangent line
Hello, I have a geometry problem that is almost killing me... I don't know how to solve it... It says... "There is a circle with radius = sqrt(2) whose center is on the xaxis, but we don't know where. We know that it has a tangent line which is y = x  1." a) Calculate the possible centers. Write the equations of the circle. b) Find the intersection and graph. So, the center would be (a,0). But... do you know how to solve it? Thank you very much!! 
November 19th, 2013, 11:12 AM  #2 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Calculate the center of a circle given a tangent line
The line perpendicular to a tangent goes thru the centre. So, you need to find points on the line y = x1 that are from the xaxis along a perpendicular line. Drawing a diagram will help. 
November 19th, 2013, 12:18 PM  #3 
Member Joined: Nov 2013 Posts: 34 Thanks: 1  Re: Calculate the center of a circle given a tangent line
Thank you!

November 19th, 2013, 12:40 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,105 Thanks: 2324 
The circle has equation (x  a)² + y² = 2, where a is to be found. As y = x  1 is a tangent line, (x  a)² + (x  1)² = 2 has a repeated root, which implies (2a + 2)² = 8(a²  1). Solving that gives a = 1 or 3. 

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calculate, center, circle, line, tangent 
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