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November 19th, 2013, 10:49 AM   #1
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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!!
lauchagonzalez is offline  
 
November 19th, 2013, 11:12 AM   #2
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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 from the x-axis along a perpendicular line.

Drawing a diagram will help.
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November 19th, 2013, 12:18 PM   #3
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Re: Calculate the center of a circle given a tangent line

Thank you!
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November 19th, 2013, 12:40 PM   #4
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The circle has equation (x - a) + y = 2, where a is to be found.
As y = x - 1 is a tangent line, (x - a) + (x - 1) = 2 has a repeated root,
which implies (2a + 2) = 8(a - 1).
Solving that gives a = -1 or 3.
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