document.write( "Question 1203378: Find the area of the region enclosed by the graph of the equation $x^2-14x+3y+70=15+9y-y^2$ that lies below the line $y=x-3$. \n" ); document.write( "
Algebra.Com's Answer #838909 by Edwin McCravy(20062) $\"\"$ $\"About$
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document.write( "First we get the equation x^2-14x+3y+70=15+9y-y^2 into standard form:\r\n" );
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document.write( "completing the square:\r\n" );
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document.write( "So it's a circle with center (7,3) and radius \r\n" );
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document.write( "Now we'll find the points of intersection with the line y = x-3:\r\n" );
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document.write( "So the points of intersection are \r\n" );
document.write( "Let's graph the circle and line:\r\n" );
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document.write( "We want to find the major segment of the circle below the line.\r\n" );
document.write( "We now see that calculus is not the way to go here.  So we must\r\n" );
document.write( "resort to geometry and analytic geometry.  We draw a radius and\r\n" );
document.write( "a perpendicular to the chord, which forms a right triangle \r\n" );
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document.write( "We find the length of the green side of the right triangle. One end is\r\n" );
document.write( "the center of the circle (7,3).  The other end is the midpoint of the\r\n" );
document.write( "chord.  We find the midpoint of the chord:\r\n" );
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document.write( "So the midpoint of the chord is \r\n" );
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document.write( "We use the distance formula to find the length of the green side\r\n" );
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document.write( "So the green leg of the right triangle is  \r\n" );
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document.write( "We use the Pythagorean theorem to find the other leg of the right triangle,\r\n" );
document.write( "The hypotenuse is the radius of the circle .\r\n" );
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document.write( "other leg = half of chord = \r\n" );
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document.write( "So the area of the right triangle is\r\n" );
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document.write( "We find the angle between the red and green lines from\r\n" );
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document.write( "We now draw in the radius to the other point of intersection of the circle and the line:\r\n" );
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document.write( "This gives us another right triangle congruent to the other one.\r\n" );
document.write( "Now the angle on the minor segment is twice the angle above or\r\n" );
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document.write( " or \r\n" );
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document.write( "To find the angle of the major segment we subtract from \r\n" );
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document.write( "The area of the sector = \r\n" );
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document.write( "Finally we add the area of the two right triangles:\r\n" );
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document.write( " square units\r\n" );
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document.write( "Edwin
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