document.write( "Question 539220: Given the circle (x - 6)2 + (y + 2)2 = 9, determine if the line with an equation of 3x - 2y = 12 is a tangent to the circle, a secant to the circle, or neither.
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Algebra.Com's Answer #353365 by AnlytcPhil(1806) $\"\"$ $\"About$

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Given the circle (x - 6)2 + (y + 2)2 = 9, determine if the line with an equation of 3x - 2y = 12 is a tangent to the circle, a secant to the circle, or neither.
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document.write( "Let's see what we believe the answer is by looking at the graph.  Then we'll show it by algebra.\r\n" );
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document.write( "Judging by the graph it looks as though the line cuts through the\r\n" );
document.write( "circle in two points, so it's probably a secant line, unless my \r\n" );
document.write( "graph is off.\r\n" );
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document.write( "We will solve the system to see how many points of intersection there is.\r\n" );
document.write( "There could be none, one or two.\r\n" );
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document.write( "(1)       (x - 6)² + (y + 2)² = 9\r\n" );
document.write( "(2)       3x - 2y = 12\r\n" );
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document.write( "We multiply the first equation out and simplify:\r\n" );
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document.write( "(1)       (x - 6)(x - 6) + (y + 2)(y + 2) = 9\r\n" );
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document.write( "(3)       x² - 12x + 36 + y² + 4y + 4 = 9\r\n" );
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document.write( "We solve (2) for y\r\n" );
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document.write( "(2)       3x - 2y = 12\r\n" );
document.write( "              -2y = -3x + 12\r\n" );
document.write( "            = x +  \r\n" );
document.write( "                y = x - 6\r\n" );
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document.write( "Substitute in (3)\r\n" );
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document.write( "(3)       x² - 12x + 36 + y² + 4y + 4 = 9\r\n" );
document.write( "       x² - 12x + 36 + (x - 6)² + 4(x - 6) + 4 = 9\r\n" );
document.write( "   x² - 12x + 36 + (x - 6)(x - 6) + 6x - 24 + 4 = 9\r\n" );
document.write( "   x² - 6x + 16 + (x² - 9x - 9x + 36) = 9\r\n" );
document.write( "   x² - 6x + 16 + x² - 9x - 9x + 36 = 9\r\n" );
document.write( "   x² - 24x + 52 + x² = 9\r\n" );
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document.write( "Multiply through by 4 to clear of fractions:\r\n" );
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document.write( "  4x² - 96x + 208 + 9x² = 36\r\n" );
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document.write( " 13x² - 96x + 208 = 36\r\n" );
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document.write( " 13x² - 96x + 172 = 0\r\n" );
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document.write( "We could solve that for x, they find y, but you weren't told to find \r\n" );
document.write( "the solution, just to determine whether it was a secant line.  So we \r\n" );
document.write( "just need to show that that equation has two real solurtions:\r\n" );
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document.write( "The discriminant is b²-4ac = (-96)² - 4(13)(172) = 9216 - 8944 = 272\r\n" );
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document.write( "That is positive.  Therefore there are two REAL solutions for x.\r\n" );
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document.write( "Therefore there are two points of intersection with the circle. So\r\n" );
document.write( "it is a secant line.\r\n" );
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document.write( "You can go on and use the quadratic formula to find the two points of\r\n" );
document.write( "intersection, but it isn't necessary.  They are, however:\r\n" );
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document.write( "(x,y) = (, )\r\n" );
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document.write( "and\r\n" );
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document.write( "(x,y) = (, )\r\n" );
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document.write( "Edwin

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