document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170335>Question 1170335</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the length of the latus rectum and the equation of the parabola with vertex at the origin, directrix x=-3 and focus (3, 0). Sketch the graph. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851214 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851214\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Determine the Orientation of the Parabola:**\r<BR>\n" );
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document.write( "* The vertex is at the origin (0, 0).<BR>\n" );
document.write( "* The focus is at (3, 0).<BR>\n" );
document.write( "* The directrix is x = -3.<BR>\n" );
document.write( "* Since the focus is to the right of the vertex and the directrix is to the left, the parabola opens to the right.\r<BR>\n" );
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document.write( "**2. Determine the Value of 'p':**\r<BR>\n" );
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document.write( "* The distance between the vertex and the focus is 'p'.<BR>\n" );
document.write( "* The distance between the vertex (0, 0) and the focus (3, 0) is 3.<BR>\n" );
document.write( "* Therefore, p = 3.\r<BR>\n" );
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document.write( "**3. Find the Equation of the Parabola:**\r<BR>\n" );
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document.write( "* Since the parabola opens to the right and the vertex is at the origin, the equation is of the form:<BR>\n" );
document.write( "   $y^2 = 4px$<BR>\n" );
document.write( "* Substitute p = 3 into the equation:<BR>\n" );
document.write( "   $y^2 = 4(3)x$<BR>\n" );
document.write( "   $y^2 = 12x$\r<BR>\n" );
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document.write( "**4. Find the Length of the Latus Rectum:**\r<BR>\n" );
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document.write( "* The length of the latus rectum is 4p.<BR>\n" );
document.write( "* Since p = 3, the length of the latus rectum is 4(3) = 12.\r<BR>\n" );
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document.write( "**5. Sketch the Graph:**\r<BR>\n" );
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document.write( "* **Vertex:** (0, 0)<BR>\n" );
document.write( "* **Focus:** (3, 0)<BR>\n" );
document.write( "* **Directrix:** x = -3<BR>\n" );
document.write( "* **Latus Rectum:** The latus rectum passes through the focus and is perpendicular to the axis of symmetry. Its endpoints are at a distance of 2p from the focus.<BR>\n" );
document.write( "   * Since 2p = 6, the endpoints of the latus rectum are at (3, 6) and (3, -6).\r<BR>\n" );
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document.write( "**Graph:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "      ^ y-axis<BR>\n" );
document.write( "      |<BR>\n" );
document.write( "    6 |      * (3, 6)<BR>\n" );
document.write( "      |     /<BR>\n" );
document.write( "      |    /<BR>\n" );
document.write( "      |   /<BR>\n" );
document.write( "      |  /<BR>\n" );
document.write( "    0 +-----------+---> x-axis<BR>\n" );
document.write( "      | \<BR>\n" );
document.write( "      |  \<BR>\n" );
document.write( "      |   \<BR>\n" );
document.write( "   -6 |      * (3, -6)<BR>\n" );
document.write( "      |<BR>\n" );
document.write( "  -3  |   Directrix x=-3<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* **Length of the Latus Rectum:** 12<BR>\n" );
document.write( "* **Equation of the Parabola:** $y^2 = 12x$<BR>\n" );
document.write( " </SPAN>\n" );
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