document.write( "Question 908274: Find the sqaure of the length of the tangent to the point (-4,1) on the circle given by the equation x^2+y^2+8x+12y+5=0. \n" ); document.write( "

Algebra.Com's Answer #550966 by Edwin McCravy(20060) $\"\"$ $\"About$

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document.write( "First we put the circle's equation, which is in GENERAL form: \r\n" );
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document.write( "in STANDARD form\r\n" );
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document.write( "Rearrange the terms so that the x term is just right of the x² term,\r\n" );
document.write( "and the y term is just right of the y² term, and the\r\n" );
document.write( "constant term is on t he right.\r\n" );
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document.write( "Now we do some work to the side to find the number necessary to \r\n" );
document.write( "complete the square on :\r\n" );
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document.write( "1. Multiply the coefficient of x, which is 8, by , getting 4.\r\n" );
document.write( "2. Square 4, getting 16.\r\n" );
document.write( "3. Add +16 just after 8x on the left and also add +16 to the right side.\r\n" );
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document.write( "Also to the side we find the number necessary to \r\n" );
document.write( "complete the square on :\r\n" );
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document.write( "1. Multiply the coefficient of y, which is 12, by , getting 6.\r\n" );
document.write( "2. Square 6, getting 36.\r\n" );
document.write( "3. Add +36 just after 12y on the left and also add +36 to the right side.\r\n" );
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document.write( "Factor the first three terms on the left as  and write as\r\n" );
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document.write( "Factor the last three terms on the left as  and write as\r\n" );
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document.write( "Combine the numbers on the right as 35.\r\n" );
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document.write( "Now we have the standard form of the circle:\r\n" );
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document.write( "Comparing to\r\n" );
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document.write( "We see that -h=+4 or h=-4, -k=+6 or k=-6, r²=47,\r\n" );
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document.write( "Now we can sketch the graph of the circle with center O(-4,-6) \r\n" );
document.write( "and radius . Wow, as we see, the point P(-4,1) is\r\n" );
document.write( "very close to the circle!. I'll draw the tangent line PT in red,\r\n" );
document.write( "and the radius to the point of tangency and the line from the \r\n" );
document.write( "center to P(4,1) in green.  We want to find the length of PT.\r\n" );
document.write( "The line OP is 7 units because it's directly above O, which is 6\r\n" );
document.write( "units below the x-axis, to P which is 1 unit above the x-axis, so\r\n" );
document.write( "OP is 6+1 or 7 units, and OT is . Triangle POT is a\r\n" );
document.write( "right triangle, so by the Pythagorean theorem:\r\n" );
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document.write( "Edwin

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