document.write( "Question 885215: A circle passes through the centre of a circle X^2 + Y^2 - 3X - 4Y - 1 = 0 and it's centre is (5,2).find it's equation. \n" ); document.write( "

Algebra.Com's Answer #535029 by KMST(5328) $\"\"$ $\"About$

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For the first circle, the given equation is $\"x%5E2%2By%5E2-3x-4y-1=0\"$ .
\n" ); document.write( "We can transform it:
\n" ); document.write( " $\"x%5E2%2By%5E2-3x-4y=1\"$
\n" ); document.write( " $\"x%5E2-3x%2By%5E2-4y=1\"$
\n" ); document.write( " $\"x%5E2-3x%2B9%2F4%2By%5E2-4y%2B4=1%2B9%2F4%2B4\"$
\n" ); document.write( " $\"%28x-3%2F2%29%5E2%2B%28y-2%29%5E2=1%2B9%2F4%2B4\"$
\n" ); document.write( "
\n" ); document.write( " $\"x=3%2F2\"$ and $\"y=2\"$ are the coordinates of the center of the circle,
\n" ); document.write( "and $\"%28x-3%2F2%29%5E2%2B%28y-2%29%5E2\"$ is the square of the distance from a point (x,y) to that center.
\n" ); document.write( " $\"1%2B9%2F4%2B4\"$ is the square of the radius of the circle, but I do not need to calculate it.
\n" ); document.write( "
\n" ); document.write( "The distance from the center of that first circle (with $\"x=3%2F2\"$ and $\"y=2\"$ )
\n" ); document.write( "to point (5,2) (also with $\"y=2\"$ ) is $\"abs%285-3%2F2%29=abs%287%2F2%29=7%2F2\"$ .
\n" ); document.write( "
\n" ); document.write( "A circle centered at (5,2), with radius $\"R\"$ has the equation
\n" ); document.write( " $\"%28x-5%29%5E2%2B%28y-2%29%5E2=R%5E2\"$
\n" ); document.write( "If the circle passes through a point with $\"x=3%2F2\"$ and $\"y=2\"$ ,
\n" ); document.write( "which is at a distance $\"7%2F2\"$ from (5,2), then $\"R=7%2F2\"$ , and
\n" ); document.write( " $\"%28x-5%29%5E2%2B%28y-2%29%5E2=%287%2F2%29%5E2\"$ is the equation of the second circle.
\n" ); document.write( "It can also be written as
\n" ); document.write( " $\"%28x-5%29%5E2%2B%28y-2%29%5E2=49%2F4\"$ , or further manipulated into whatever equivalent equation you desire:
\n" ); document.write( " $\"x%5E2-10x%2B25%2By%5E2-4y%2B4=49%2F4\"$ --> $\"x%5E2%2By%5E2-10x-4y%2B25%2B4-49%2F4=0\"$ --> $\"x%5E2%2By%5E2-10x-4y%2B67%2F4=0\"$ --> $\"4x%5E2%2B4y%5E2-40x-16y%2B67=0\"$ \n" ); document.write( "

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