document.write( "Question 1146372: Find the equation of the radical axis of the two circles x2 + y2 - 4x - 6y - 3 = 0 and x2 + y2 - 12x - 14y + 65 = 0. \n" ); document.write( "

Algebra.Com's Answer #854708 by KMST(5398) $\"\"$ $\"About$

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The radical axis of two circles can be defined as the locus (set) of the points from where tangents to the two circles would have equal lengths.
\n" ); document.write( "It can also be defined as the locus of the points whose power relative to both circles is the same.
\n" ); document.write( "Power of a point relative to a circle is $\"d%5E2-r%5E2\"$ , where
\n" ); document.write( " $\"d=distance\"$ $\"to\"$ $\"the\"$ $\"circle\"$ $\"center\"$ , and
\n" ); document.write( " $\"r=radius\"$ $\"of\"$ $\"the\"$ $\"circle\"$ .
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\n" ); document.write( "The definitions are equivalent, because in the right triangles formed by tangent, the radius, and the distance $\"d\"$ .
\n" ); document.write( "The line segment of length $\"d\"$ between the point and the circle center is the hypotenuse and the power of the point is the square of the length of the tangent segment,
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\n" ); document.write( "UNDERSTANDING THE PROBLEM:
\n" ); document.write( "The equations of the circles are written in the general form.
\n" ); document.write( "They could be written in a form that allowed us to identify the center and radius of each circle.
\n" ); document.write( "For the first circle, that would be $\"%28x-2%29%5E2%2B%28y-3%29%5E2-%283%2B2%5E2%2B3%5E2%29=0\"$ <--> $\"%28x-2%29%5E2%2B%28y-3%29%5E2-16=0\"$ , showing the center to be C(2,3) , and the radius to be $\"sqrt%2816%29=4\"$ .
\n" ); document.write( "For the second circle, that would be $\"%28x-6%29%5E2%2B%28y-7%29%5E2-%286%5E2%2B7%5E2-65%29=0\"$ <--> $\"%28x-6%29%5E2%2B%28y-7%29%5E2-20=0\"$ , showing the center to be C'(6,7) , and the radius to be $\"sqrt%2820%29\"$ .
\n" ); document.write( "The equations in this form show a sum of squares that is the square of the distance from a point P(x,y) to the center of the circle, minus the square of the radius.
\n" ); document.write( "That is the power of point P(x,y) relative to the circle.
\n" ); document.write( "For points of a circle it is zero, their power relative to the circle is zero,
\n" ); document.write( "but for any point P(x,y) the power of P relative to the first circle can be written as
\n" ); document.write( " $\"%28x-2%29%5E2%2B%28y-3%29%5E2-16\"$ , or as its equivalent form, $\"x%5E2%2By%5E2-4x-6y-3\"$ .
\n" ); document.write( "So, for all the points P(x,y) whose powers relative to both circles are the same,
\n" ); document.write( " $\"x%5E2%2By%5E2-4x-6y-3=x%5E2%2By%5E2-12x-14y%2B65\"$ , and from there we get the equation of the radical axis of the two circles.
\n" ); document.write( " $\"cross%28x%5E2%2By%5E2%29-4x-6y-3=cross%28x%5E2%2By%5E2%29-12x-14y%2B65\"$ --> $\"-4x%2B12x-6y%2B14y=65%2B3\"$ --> $\"8x%2B8y=68\"$ --> $\"x%2By=68%2F8\"$ --> $\"highlight%28y=8.5-x%29\"$
\n" ); document.write( "The circles, with their centers, the line segment connecting those centers, and the radical axis are shown below.
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