document.write( "Question 1205247: 1.Find the equation of the circle that passes through the points (0,0), (5,0), and (3,3). Use both geometric and algebraic solution for comparison.\r
\n" ); document.write( "\n" ); document.write( "2. The cable of a suspension bridge hangs in the shape of a parabola. The tower supporting the cable are 400 ft apart and 150 ft high. If the cable, at its lowest, is 30 ft above the bridge at its midpoint, how high is the cable 50 ft away(horizontally) from either tower? \n" ); document.write( "
Algebra.Com's Answer #841926 by greenestamps(13203)\"\" \"About 
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\n" ); document.write( "I don't know what it means to use a geometric solution to find the equation of a circle. At some point, no matter how much work you do with geometry, you need to use algebra to get an equation.

\n" ); document.write( "The perpendicular bisector of any chord of a circle passes through the center of the circle.

\n" ); document.write( "One chord of the given circle is the segment with endpoints (0,0) and (5,0). That is a horizontal segment with midpoint (0,2.5); the perpendicular bisector of that segment is the vertical line x=2.5. So the center of the circle lies somewhere on the line x=2.5; the coordinates of the center of the circle are (2.5,y), where y is to be determined.

\n" ); document.write( "For a purely algebraic method for finding the center, we can use the fact that the distance from the center to any point on the circle is constant, so the distance from (2.5,y) to (5,0) is the same as the distance from (2.5,y) to (3,3):

\n" ); document.write( "\"%282.5%29%5E2%2By%5E2=%280.5%29%5E2%2B%28y-3%29%5E2\"
\n" ); document.write( "\"6.25%2By%5E2=0.25%2By%5E2-6y%2B9\"
\n" ); document.write( "\"6y=9.25-6.25=3\"
\n" ); document.write( "\"y=0.5\"

\n" ); document.write( "The center of the circle is (2.5,0.5).

\n" ); document.write( "Use that center point and any of the three given points on the circle to find that the radius of the circle is \"sqrt%286.5%29\". Then plug the coordinates of the center and the radius in the standard form for the equation of a circle to find the equation is

\n" ); document.write( "\"%28x-2.5%29%5E2%2B%28y-0.5%29%5E2=6.5\"

\n" ); document.write( "An alternative algebraic method for finding the center is by finding the intersection of any two chords of the circle.

\n" ); document.write( "We have the equation of one chord: x=2.5.

\n" ); document.write( "A little algebra shows that the perpendicular bisector of the chord joining (0,0) and (3,3) has the equation y=-x+3; the intersection of those two chords is (2.5,0.5).

\n" ); document.write( "I leave the details of those calculations to the student.

\n" ); document.write( "ANSWER: \"%28x-2.5%29%5E2%2B%28y-0.5%29%5E2=6.5\"

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