document.write( "Question 1183865: Find the equation of the locus of a point which moves so that its distance from the point (1,-1) is three times its distance from the line y = 3. \n" ); document.write( "
Algebra.Com's Answer #814359 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Find the equation of the locus of a point which moves so that its distance
\n" ); document.write( "from the point (1,-1) is three times its distance from the line y = 3.
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\n" ); document.write( "\n" ); document.write( "            The setup equation is  INCORRECT  in the post by @josgaritmetic.\r
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\n" ); document.write( "\n" ); document.write( "            THEREFORE,  his  \"solution\"  is  WRONG  from the very first line to very last line.\r
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\n" ); document.write( "\n" ); document.write( "            For your safety,  ignore his post.\r
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\n" ); document.write( "\n" ); document.write( "            I came to bring a correct solution.\r
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document.write( "Write the equation for distances, as you read the problem\r\n" );
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document.write( "    \"sqrt%28%28x-1%29%5E2+%2B+%28y-+%28-1%29%29%5E2%29\" = 3*|y-3|, \r\n" );
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document.write( "or\r\n" );
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document.write( "    \"sqrt%28%28x-1%29%5E2+%2B+%28y%2B1%29%5E2%29\" = 3*|y-3|.\r\n" );
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document.write( "Square both sides\r\n" );
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document.write( "    (x-1)^2 + (y+1)^2 = 9*(y-3)^2.\r\n" );
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document.write( "Simplify and reduce to the standard conic section equation\r\n" );
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document.write( "    (x-1)^2 +  y^2 + 2y + 1 = 9y^2 - 54y + 81\r\n" );
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document.write( "    (x-1)^2 - 8y^2 + 56y + 1  = 81\r\n" );
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document.write( "    (x-1)^2 - 8(y^2 - 7y) + 1 = 81\r\n" );
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document.write( "    (x-1)^2 - 8(y^2 - 2*3.5y + 3.5^2) + 8*3.5^2 + 1 = 81\r\n" );
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document.write( "    (x-1)^2 - 8(y-3.5)^2 = 81 - 8*3.5^2 - 1\r\n" );
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document.write( "    (x-1)^2 - 8(y-3.5)^2 = - 18\r\n" );
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document.write( "    8(y-3.5)^2 - (x-1)^2 = 18\r\n" );
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document.write( "    \"%28y-3.5%29%5E2%2F%28%289%2F4%29%29\" - \"%28x-1%29%5E2%2F18\" = 1.\r\n" );
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document.write( "The last equation describes a hyperbola with the center at the point (1,3.5),\r\n" );
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document.write( "vertical real semi-axis of the length  \"sqrt%289%2F4%29\" = \"3%2F2\"  (tranverse semi-axis),\r\n" );
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document.write( "and horizontal imaginary semi-axis of the length  \"sqrt%2818%29\" = \"3%2Asqrt%282%29\".\r\n" );
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document.write( "The hyperbola is open vertically up and down.\r\n" );
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\n" ); document.write( "\n" ); document.write( "For hyperbola, its canonical equation, standard form equation, general equation, elements and properties,\r
\n" ); document.write( "\n" ); document.write( "see the lessons\r
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\n" ); document.write( "\n" ); document.write( "    - Hyperbola definition, canonical equation, characteristic points and elements \r
\n" ); document.write( "\n" ); document.write( "    - Hyperbola focal property \r
\n" ); document.write( "\n" ); document.write( "    - Tangent lines and normal vectors to a hyperbola \r
\n" ); document.write( "\n" ); document.write( "    - Optical property of a hyperbola \r
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\n" ); document.write( "\n" ); document.write( "    - Standard equation of a hyperbola \r
\n" ); document.write( "\n" ); document.write( "    - Identify elements of hyperbola given by its standard equation \r
\n" ); document.write( "\n" ); document.write( "    - Find the standard equation of a hyperbola given by its elements \r
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\n" ); document.write( "\n" ); document.write( "    - General equation of a hyperbola\r
\n" ); document.write( "\n" ); document.write( "    - Transform general equation of a hyperbola to the standard form by completing the square \r
\n" ); document.write( "\n" ); document.write( "    - Identify elements of a hyperbola given by its general equation \r
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\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on hyperbolas \r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this online textbook under the topic
\n" ); document.write( "\"Conic sections: Hyperbolas. Definition, major elements and properties. Solved problems\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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\n" ); document.write( "\n" ); document.write( "After seeing my post,  @josgarithmetic rewrote his setup exactly as my,  so now his post is safe  (although useless).\r
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\n" ); document.write( "\n" ); document.write( "At least,  I forced him to make it safe . . . \r
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