Question 1084921
Determine the slope of the line from (4,2) to (2,-1).
{{{m=(2-(-1))/(4-2)=3/2}}}
Now determine the slope of the bisector since they're perpendicular,
{{{m[1]*m[2]=-1}}}
{{{(3/2)m[2]=-1}}}
{{{m[2]=-2/3}}}
Determine the equation of the line using the point and the slope,
{{{y-(-1)=(-2/3)(x-2)}}}
{{{y+1=(2/3)x+4/3}}}
{{{y=-(2/3)x+1/3}}}
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*[illustration CD25.JPG].
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Now find the intersection of that line with the circle.
{{{(x-4)^2+((1-2x)/3-2)^2=25}}}
{{{x^2-8x+16+(4x^2+20x+25)/9=25}}}
{{{9x^2-72x+144+4x^2+20x+25=225}}}
{{{13x^2-52x+169=225}}}
{{{13(x^2-4x+13)=225}}}
{{{x^2-4x+13=225/13}}}
{{{(x^2-4x+4)+9=225/13}}}
{{{(x-2)^2=225/13-117/13}}}
{{{(x-2)^2=108/13}}}
{{{x-2=0 +- sqrt(108/13)}}}
{{{x=2 +- sqrt(108/13)}}}
{{{x=2 +- (6/13)sqrt(39)}}}
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*[illustration CD26.JPG].
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So then,
{{{y=-1 +- (4/13)sqrt(39)}}}
Finally calculate the distance from the two intersection points using the distance formula, that's the length of the chord.
{{{D^2=( 2+(6/13)sqrt(39)-(2-(6/13)sqrt(39)) )^2+( -1+(4/13)sqrt(39))-(-1-(4/13)sqrt(39) )^2}}}
{{{D^2=432/13 +192/13}}}
{{{D^2=624/13}}}
{{{D^2=48}}}
{{{D=sqrt(48)}}}
{{{D=4sqrt(3)}}}