Algebra.Com's Answer #664229 by ikleyn(52788)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Determine the value(s) of k such that the circle x^2+(y-6)^2 = 36 and the parabola x^2 = 4ky will intersect only at the origin.
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document.write( " = 36, (1)\r\n" );
document.write( " = 4ky (2)\r\n" );
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document.write( "The circle =  has the center at (x,y) = (0,6) and has the radius of 6. \r\n" );
document.write( "So, the circle has the y-axis x=0 as a diameter and as a symmetry line, passes through the origin and touches the x-axis.\r\n" );
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document.write( "Parabola =  also passes through the origin; has the y-axis x=0 as its symmetry line, and touches the x-axis.\r\n" );
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document.write( "After these geometric considerations (that are useful but are not absolutely necessary) let solve the problem algebraically.\r\n" );
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document.write( "Based on (2), substitute 4ky instead of into the equation (1). You will get\r\n" );
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document.write( " = 36.\r\n" );
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document.write( "So, in this way you excluded \"x\" from the system and got a single equation for \"y\". Let us simplify it:\r\n" );
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document.write( " = 36, or\r\n" );
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document.write( " = 0. (1)\r\n" );
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document.write( "Now, the problem requires this equation (1) to have only one non-negative solution.\r\n" );
document.write( " (One solution is evident/obvious. It is y = 0.)\r\n" );
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document.write( "It implies that (4k-12) MUST be non-positive: 4k-12 <= 0.\r\n" );
document.write( "OTHERWISE y =  would be the other non-negative solution to (1).\r\n" );
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document.write( "So, the solution to the problem is this inequality 4k-12 <= 0, or, equivalently, k <= 3 (3 =  )\r\n" );
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document.write( "Answer. k <= 3.\r\n" );
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document.write( "See an illustration below for k = 3, 2, and 4.\r
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document.write( "The circle = (red + green)
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document.write( "and three parabolas x^2 = 4ky for k=3 (blue), k=2, and k=4. \r
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document.write( "The circle = (green)
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document.write( "and three parabolas x^2 = 4ky for k=3 (blue), k=2, and k=4. \r
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document.write( "For solution of similar problems see the lesson\r
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document.write( " - Solving systems of algebraic equations of degree 2 \r
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document.write( "in this site.\r
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document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
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document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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