document.write( "Question 554469: hi, i need your help please... i don't know how to answer these questions because i can't fully imagine them... i hope you can help me because i really want to understand this lesson for the sake of my grades.....\r
\n" ); document.write( "\n" ); document.write( "a) the sum of the distance from a point P to (4,0) and (-4,0) is 9. if the abscissa of P is 1, find its ordinate...\r
\n" ); document.write( "\n" ); document.write( "b)the center of a circle is at (-3,-2). if a chord of length 4 is bisected at (3,1), find the length of the radius...\r
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Algebra.Com's Answer #852362 by n2(2) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "(a) the sum of the distance from a point P to (4,0) and (-4,0) is 9. if the abscissa of P is 1, find its ordinate.
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\n" ); document.write( "\n" ); document.write( "        I am @ikleyn. Hello again.\r
\n" ); document.write( "\n" ); document.write( "        This 'n2' is my second nickname, which I created to place here my solution to part (a) of the problem.\r
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\n" ); document.write( "\n" ); document.write( "        Tutor @greenestamps solved this part nicely by applying theory of ellipses.\r
\n" ); document.write( "\n" ); document.write( "        In my solution below, I work in the frame of knowledge related to triangle geometry, only.\r
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document.write( "Let A, C and P be the given points A = (-4,0), C = (4,0).\r\n" );
document.write( "Let P = (1,y) be the point, for which we want to find its coordinate 'y' such that\r\n" );
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document.write( "    |AP| + |CP| = 9.    (1)\r\n" );
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document.write( "Notice that line AC is horizontal (lies on x-axis of the (x,y) coordinate system).\r\n" );
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document.write( "Draw a perpendicular PD from vertex P of triangle APC to its base AC, \r\n" );
document.write( "so point D = (1,0) is the intersection of the perpendicular with AC.\r\n" );
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document.write( "Thus we have now triangle APC and two right-angled triangles ADP and CDP.\r\n" );
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document.write( "Let 'a' be the length CP and 'c' be the length AP:  \r\n" );
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document.write( "    a = |CP|,  c = |AP|.    (2)\r\n" );
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document.write( "We have  |AD| = 1 - (-4) = 5;  |CD| = 4 - 1 = 3.   (3)\r\n" );
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document.write( "Perpendicular PD is the common leg of triangles ADP and CDP, so we can write, using Pythagorean equation \r\n" );
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document.write( "   |AP|^2 - |AD|^2 = |CP|^2 - |CD|^2.\r\n" );
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document.write( "Substituting here from relations (2) and (3), we get this equation\r\n" );
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document.write( "   c^2 - 5^2 = a^2 - 3^2,\r\n" );
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document.write( "which implies \r\n" );
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document.write( "    c^2 - a^2 = 5^2 - 3^2,\r\n" );
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document.write( "    c^2 - a^2 = 16.    (4)\r\n" );
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document.write( "So, now we have this system of equations (1) and (4)\r\n" );
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document.write( "    c   + a   =  9.    (1')\r\n" );
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document.write( "    c^2 - a^2 = 16,    (4')\r\n" );
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document.write( "Now we are on a finish line to complete the solution.\r\n" );
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document.write( "In equation (4'),  factor left side as  c^2 - a^2 = (c+a)*(c-a) and replace (c+a) by 9,\r\n" );
document.write( "based on equation (1').  Then instead the system (1'), (4') you will get the system\r\n" );
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document.write( "    a  + c =  9.    (1'')\r\n" );
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document.write( "    9(c-a) = 16.    (4'')\r\n" );
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document.write( "Open parentheses in (4'')  and multiply equation (1'')  by 9  (both sides)\r\n" );
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document.write( "    9c + 9a = 81.\r\n" );
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document.write( "    9c - 9a = 16,\r\n" );
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document.write( "Add      the two last equations and get  18c = 81 + 16 = 97,  c = 97/18.\r\n" );
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document.write( "Subtract the two last equations and get  18a = 81 - 16 = 65,  c = 65/18.\r\n" );
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document.write( "Now we can find 'y'\r\n" );
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document.write( "    y^2 = |PD|^2 = |CP|^2 - |AD|^2 = a^2 - 3^2 =  =  = ,\r\n" );
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document.write( "    y =  = 2.010005835,  or  y = 2.01, approximately.\r\n" );
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document.write( "ANSWER.  y =  = 2.010005835,  or  y = 2.01, approximately.\r\n" );
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