document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170336>Question 1170336</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the focus, directrix, and equation of the parabola with vertex at the origin, axis along y-axis; opening upward; and the length of the latus rectum is 12. 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 #851213 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=851213\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Equation of the Parabola:**\r<BR>\n" );
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document.write( "* Since the vertex is at the origin (0, 0), the axis is along the y-axis, and the parabola opens upward, the equation of the parabola is of the form:<BR>\n" );
document.write( "   $x^2 = 4py$<BR>\n" );
document.write( "   where $p > 0$.\r<BR>\n" );
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document.write( "**2. Length of the Latus Rectum:**\r<BR>\n" );
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document.write( "* The length of the latus rectum is given as 12.<BR>\n" );
document.write( "* The length of the latus rectum is also equal to $4p$.<BR>\n" );
document.write( "* Therefore, $4p = 12$.<BR>\n" );
document.write( "* Solving for $p$:<BR>\n" );
document.write( "   $p = \frac{12}{4} = 3$\r<BR>\n" );
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document.write( "**3. Equation of the Parabola (Specific):**\r<BR>\n" );
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document.write( "* Substitute $p = 3$ into the equation $x^2 = 4py$:<BR>\n" );
document.write( "   $x^2 = 4(3)y$<BR>\n" );
document.write( "   $x^2 = 12y$\r<BR>\n" );
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document.write( "**4. Focus:**\r<BR>\n" );
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document.write( "* The focus of the parabola is at (0, p).<BR>\n" );
document.write( "* Since $p = 3$, the focus is at (0, 3).\r<BR>\n" );
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document.write( "**5. Directrix:**\r<BR>\n" );
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document.write( "* The directrix is a horizontal line given by the equation $y = -p$.<BR>\n" );
document.write( "* Since $p = 3$, the directrix is $y = -3$.\r<BR>\n" );
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document.write( "**6. Sketch the Graph:**\r<BR>\n" );
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document.write( "* **Vertex:** (0, 0)<BR>\n" );
document.write( "* **Focus:** (0, 3)<BR>\n" );
document.write( "* **Directrix:** y = -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 (-6, 3) and (6, 3).\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( "      |<BR>\n" );
document.write( "    3 |   * (0, 3) - Focus<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( "   -3 |   v - Directrix<BR>\n" );
document.write( "      |<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* **Equation of the Parabola:** $x^2 = 12y$<BR>\n" );
document.write( "* **Focus:** (0, 3)<BR>\n" );
document.write( "* **Directrix:** $y = -3$<BR>\n" );
document.write( " </SPAN>\n" );
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