SOLUTION: the center of a circle is at (4,2) and its radius is 5. find the length of the chord which is bisected at (2,-1).

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Question 1084921: the center of a circle is at (4,2) and its radius is 5. find the length of the chord which is bisected at (2,-1).
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Determine the slope of the line from (4,2) to (2,-1).
m=%282-%28-1%29%29%2F%284-2%29=3%2F2
Now determine the slope of the bisector since they're perpendicular,
m%5B1%5D%2Am%5B2%5D=-1
%283%2F2%29m%5B2%5D=-1
m%5B2%5D=-2%2F3
Determine the equation of the line using the point and the slope,
y-%28-1%29=%28-2%2F3%29%28x-2%29
y%2B1=%282%2F3%29x%2B4%2F3
y=-%282%2F3%29x%2B1%2F3
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Now find the intersection of that line with the circle.
%28x-4%29%5E2%2B%28%281-2x%29%2F3-2%29%5E2=25
x%5E2-8x%2B16%2B%284x%5E2%2B20x%2B25%29%2F9=25
9x%5E2-72x%2B144%2B4x%5E2%2B20x%2B25=225
13x%5E2-52x%2B169=225
13%28x%5E2-4x%2B13%29=225
x%5E2-4x%2B13=225%2F13
%28x%5E2-4x%2B4%29%2B9=225%2F13
%28x-2%29%5E2=225%2F13-117%2F13
%28x-2%29%5E2=108%2F13
x-2=0+%2B-+sqrt%28108%2F13%29
x=2+%2B-+sqrt%28108%2F13%29
x=2+%2B-+%286%2F13%29sqrt%2839%29
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So then,
y=-1+%2B-+%284%2F13%29sqrt%2839%29
Finally calculate the distance from the two intersection points using the distance formula, that's the length of the chord.

D%5E2=432%2F13+%2B192%2F13
D%5E2=624%2F13
D%5E2=48
D=sqrt%2848%29
D=4sqrt%283%29

Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
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the center of a circle is at (4,2) and its radius is 5. find the length of the chord which is bisected at (2,-1).
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Below I placed much simpler solution. 

1.  Make a sketch to follow my arguments.

    Let the point O = (4,2) be the center of the circle,
    and let the point A = (2,-1) bisects our chord.

    Let B  and C be the endpoints of the chord.


2.  Then abs%28OC%29%5E2 = abs%28OA%29%5E2 + abs%28AC%29%5E2,   or

    For |OC| you have  abs%28OC%29%5E2 = 25  and  for  |OA| you have 

     abs%28OA%29%5E2 = %284-2%29%5E2+%2B+%282-%28-1%29%29%5E2 = 2%5E2+%2B+3%5E2 = 13.


    Therefore,  abs%28AC%29%5E2 = 5%5E2+-+13 = 12.

    Then |AC| = sqrt%2812%29 = 2%2Asqrt%283%29,

         and  |BC| = 4%2Asqrt%283%29.

Answer. The length of the chord is 4%2Asqrt%283%29.