document.write( "Question 75019: Find the equation of the parabola whose focus is (2,3) and directrix is x=12. \n" ); document.write( "
Algebra.Com's Answer #53956 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
Find the equation of the parabola whose\r\n" );
document.write( "focus is (2,3) and directrix is x=12\r\n" );
document.write( "\r\n" );
document.write( "Let's plot the focus (2,3), marked with a <\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now let's draw the directrix, which is a\r\n" );
document.write( "vertical line whose equation is x=12, 12\r\n" );
document.write( "units right of the y-axis.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We can now tell that the parabola opens to \r\n" );
document.write( "the left, and has equation of the form\r\n" );
document.write( "\r\n" );
document.write( "(y - k)² = 4p(x - h)\r\n" );
document.write( "\r\n" );
document.write( "where (h, k) is the vertex, and p = the distance\r\n" );
document.write( "between the focus and the vertex, which is also\r\n" );
document.write( "the distance from the vertex to the directrix.\r\n" );
document.write( "However this distance is taken as NEGATIVE since\r\n" );
document.write( "the parabola must open to the left.\r\n" );
document.write( "\r\n" );
document.write( "The vertex is the point which is half-way between the\r\n" );
document.write( "focus and the directrix. Since the distance from the \r\n" );
document.write( "focus to the directrix is 9 units to the right, the \r\n" );
document.write( "vertex is located half of 9 units, or 4.5 units to \r\n" );
document.write( "the right of the focus. That point, which is 4.5 units \r\n" );
document.write( "to the right of the focus (2,3), is the point \r\n" );
document.write( "(2+4.5, 3) or (6.5, 3). So that's the vertex\r\n" );
document.write( "so (h, k) = (6.5,3)\r\n" );
document.write( "\r\n" );
document.write( "Now we plot the vertex with the mark =  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The only thing left to find is p, which is -4.5. It\r\n" );
document.write( "has absolute value 4.5 because that is the distance \r\n" );
document.write( "from the focus to the vertex, and also the distance\r\n" );
document.write( "from the vertex to the directrix.  And since the\r\n" );
document.write( "parabola opens left, we make it negative, so\r\n" );
document.write( "p = -4.6\r\n" );
document.write( "\r\n" );
document.write( "So now we substitute (h, k) = (6.5, 3) and p = -4.5\r\n" );
document.write( "\r\n" );
document.write( "into\r\n" );
document.write( "\r\n" );
document.write( "(y - k)² = 4p(x - h)\r\n" );
document.write( "\r\n" );
document.write( "and get\r\n" );
document.write( "\r\n" );
document.write( "(y - 3)² = 4(-4.5)(x - 6.5)\r\n" );
document.write( "\r\n" );
document.write( "(y - 3)² = -6(x - 6.5)\r\n" );
document.write( "\r\n" );
document.write( "and the graph is\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );