document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=2049>Question 2049</A>: <I><SPAN STYLE=\"word-break: break-all;\"> These are the hardest questions, I cannot figure them out. Can you help me?\r<BR>\n" );
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document.write( "What is the vertex ot the parabola with the equation (y - 1)2 = 4x?\r<BR>\n" );
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document.write( "What is the focus of the parabola with the equation (x - 1)2 + 32= 8y?\r<BR>\n" );
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document.write( "Find the center of a circle with the equation x2 + y2 + 2x + 4y - 9 = 0.\r<BR>\n" );
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document.write( "Find the radius of a circle with the equation x2 + y2 - 6x - 12y + 36 = 0.\r<BR>\n" );
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document.write( "Find the center of an ellipse with the equation 9x2 + 16y2 - 18x + 64y = 71.\r<BR>\n" );
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document.write( "Find the foci of an ellipse with the equation 7(x - 2)2 + 3(y - 2)2 = 21. \r<BR>\n" );
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document.write( "Find the length of the major and minor axes of an ellipse with the equation 16x2 + 25y2 + 32x - 150y = 159.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  These are basic questions about conic and not hard at all.<BR>\n" );
document.write( " Also,you should know how to type the square or power.\r<BR>\n" );
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document.write( " 1.What is the vertex ot the parabola with the equation (y - 1)^2 = 4x ?<BR>\n" );
document.write( "   Sol:  The equations as (y - k)^2 = c(x -h) or c(y - k) = (x -h)^2<BR>\n" );
document.write( "         are parabolas with vertex (h,k).<BR>\n" );
document.write( "         For given (y - 1)^2 = 4x = 4(x-0) , the vertex is (0,1)\r<BR>\n" );
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document.write( "2.What is the focus of the parabola with the equation (x - 1)^2 + 32= 8y? <BR>\n" );
document.write( "  Sol: As in (1), (x - 1)^2 + 32= 8y can be converted to<BR>\n" );
document.write( "       (x - 1)^2 = 8(y -4), so its vertex is (1,4).\r<BR>\n" );
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document.write( "3.Find the center of a circle with the equation x^2 + y^2 + 2x + 4y - 9 = 0. <BR>\n" );
document.write( "  Sol: Make complete square: the equation converts to<BR>\n" );
document.write( "       x^2 + 2x + 1 + y^2  + 4y + 4 = 9+1+4 = 14, or<BR>\n" );
document.write( "       (x+1)^2 + (y+2)^2 = 14. So, the center of the circle is (-1,-2)<BR>\n" );
document.write( "       and the radius is sqrt(14).<BR>\n" );
document.write( "4. Find the radius of a circle with the equation x2 + y2 - 6x - 12y + 36 = 0. <BR>\n" );
document.write( "  Sol: As in (3):x^2 - 6x + (6/2)^2 + y2  - 12y + (12/2)^2 = -36 + 9 + 36 =9.<BR>\n" );
document.write( "       Or (x -3)^2 + (y-6)^2 = 9.<BR>\n" );
document.write( "       So, the center of the circle is (3,6)<BR>\n" );
document.write( "       and the radius is 3.<BR>\n" );
document.write( "5.Find the center of an ellipse with the equation 9x^2 +16y^2 -18x+64y= 71. <BR>\n" );
document.write( "Sol: Make complete square: the equation converts to<BR>\n" );
document.write( "       9(x^2 - 2x + 1) + 16(y^2  + 4y + 4) = 71 + 9 + 64 = 144, or<BR>\n" );
document.write( "       9(x-1)^2 + 16(y + 2)^2 = 144.  Dividing each term by 144, we have:<BR>\n" );
document.write( "       (x-1)^2/16 + (y + 2)^2/9 = 1. <BR>\n" );
document.write( "     The center of the ellipse is (1,-2). <BR>\n" );
document.write( "    [also, the length of the major axis = 4 and the length of the minor axis<BR>\n" );
document.write( "     = 3]\r<BR>\n" );
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document.write( "6.Find the foci of an ellipse with the equation 7(x - 2)^2 + 3(y - 2)^2 = 21. <BR>\n" );
document.write( "  Sol: Dividing each term by 21, we have:<BR>\n" );
document.write( "       (x-1)^2/3 + (y - 2)^2/7 = 1. <BR>\n" );
document.write( "We see the center is (1,2) and the length of the major axis a = sqrt(7),<BR>\n" );
document.write( "the length of the minor axis b = sqrt(3). Note, the  long axis is along <BR>\n" );
document.write( "y-direction(vertical).<BR>\n" );
document.write( "Since c^2 = a^2-b^2 = 7-3 =4, we get c = 2.<BR>\n" );
document.write( " Hence, the foci are (1,2+c) = (1,4) and (1,2-c) = (1,0)\r<BR>\n" );
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document.write( "7.Find the length of the major and minor axes of an ellipse with the equation  Sol: As in (5), the equation converts to<BR>\n" );
document.write( "       16(x^2 + 2x + 1) + 25(y^2  - 6y + 9) = 159 +16 +225 = 400 or<BR>\n" );
document.write( "       4(x+1)^2 + 25(y-3)^2 = 200. <BR>\n" );
document.write( "       Dividing each term by 200, we have:<BR>\n" );
document.write( "       (x+1)^2/50 + (y - 2)^2/8 = 1. <BR>\n" );
document.write( "       Hence, the major axis a = sqrtt(50) = 5 sqrt(2),<BR>\n" );
document.write( "        the minor axis b = sqrtt(8) = 2 sqrt(2).\r<BR>\n" );
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document.write( " Kenny\r<BR>\n" );
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