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April 4th, 2013, 03:13 AM   #1
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Circle problems

1 a circle is tangent to the two axes and the line 4x+3y-12=0. Find it's equation.
2 consider the circle x^2 + y^2 - 277 = 0. If M(3,-5) is the midpoint of a chord of the circle, find the equation of the chord.
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April 4th, 2013, 06:08 AM   #2
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Re: Circle problems

Hello, justusphung!

Did you make any sketches?


Quote:

Use the scroll bar to see the entire diagram.

Code:
      |
    4 o
      | o
      |   o
      |     o
      |       o
      |         o
      |           o
      |       * * * o
      |   *           o
      | *               o
      |*              /  *o
      |             /r      o
      *         C /       *   o
      * - - - - o         *     o
      *    r    :         *       o
      |         :                   o
      |*        :r       *            o
      | *       :       *               o
      |   *     :     *                   o
   ---+-------*-*-*-------------------------o----
      |                                     3



There is a sneaky solution to this problem.












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April 4th, 2013, 06:33 AM   #3
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Re: Circle problems

[color=#000000]Soroban finished it first, but since I made a figure I will upload my solution.

1. Draw the line 4x+3y-12=0, you will notice the right angle triangle with its two perpendicular sides lying on the x and y axis like seen in the next figure.

[attachment=0:1j41gfsh]justusphung.png[/attachment:1j41gfsh]

Since the x-axis, y-axis and the hypoteneuse of the triangle ABC are tangent to the circle, we deduce that the circle we seek is the circle inscibed in the triangle ABC.

So the center of the circle is and since the distance of (1,1) from the x and y axis is 1 the radius we seek is 1. So the circle is

2. We are given the circle , which has center the origin of the axis (0,0). The radius which meets the chord at point M perpendicularly has equation of the form since it passes from the origin (0,0). In order to find we plug in the coordinates of the point M, so and the equation is . Now this line must be perpendicular to the line which is defined by the chord at point M. The line we seek has equation , we know that and for , the equation we seek is .[/color]
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File Type: png justusphung.png (28.5 KB, 213 views)
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April 4th, 2013, 07:15 AM   #4
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Re: Circle problems

Thank you for your reply and the precise drawings. By the way, despite of the answer you provided in question 1, there are another 3 possible answers. Can you guys help me find out the other 3 outcomes. Thanks a lot.

For question 2, ZardoZ has solved it out. reli thx.
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April 4th, 2013, 07:46 AM   #5
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Re: Circle problems

Arr! Finally, I figured it out. As there is one possible answer in section 2 and section 4 of the coordinate plane respectively, while in section 1 there are 2 possible answers, we can derive the equation by determining the distance of the circle centre to the given line, 4x+3y-12=0.
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April 4th, 2013, 07:56 AM   #6
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Re: Circle problems

[color=#000000]One other possible way to do it is to use your drawing skills, take a squared paper draw the x and y axis, then the given line and then the bisectors of all the angles of the triangle that soroban and I have mentioned in our solutions. The point where the angle bisectors meet is the centre of the circle you seek.[/color]
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