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 November 19th, 2013, 09:49 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 x-axis, 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, 10:12 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 = x-1 that are $\sqrt{2}$ from the x-axis along a perpendicular line. Drawing a diagram will help.
 November 19th, 2013, 11:18 AM #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, 11:40 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2217 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.

 Tags calculate, center, circle, line, tangent

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# how to find the center of a circle with tangent lines

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