\r\n" );
document.write( "1. Find the Equation for the parabola with focus\r\n" );
document.write( " (3,2) and directrix y=6. \r\n" );
document.write( "\r\n" );
document.write( "The focus is a point and the directrix is a line.\r\n" );
document.write( "\r\n" );
document.write( "The parabola has the equation\r\n" );
document.write( "\r\n" );
document.write( " (x - h)² = 4p(y - k)\r\n" );
document.write( "\r\n" );
document.write( "where (h,k) is the vertex, p is the distance from\r\n" );
document.write( "the vertex to the focus and also the distance\r\n" );
document.write( "from the vertex to the focus. \r\n" );
document.write( "\r\n" );
document.write( "Lets draw the focus F(3,2). I'll put an \"o\" there.\r\n" );
document.write( "And I'll draw the directrix y=6 which is a horizontal\r\n" );
document.write( "line 6 units above the x-axis. I'll draw it green\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "The vertex is halfway between the focus and the\r\n" );
document.write( "directrix. So the vertex is at the point V(3,4).\r\n" );
document.write( "\r\n" );
document.write( "I'll plot the vertex with an o.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "That means that the vertex (h,k) = (3,4)\r\n" );
document.write( "\r\n" );
document.write( "So in the equation \r\n" );
document.write( "\r\n" );
document.write( " (x - h)² = 4p(y - k)\r\n" );
document.write( "\r\n" );
document.write( "we can now put in h and k:\r\n" );
document.write( "\r\n" );
document.write( " (x - 3)² = 4p(y - 4)\r\n" );
document.write( "\r\n" );
document.write( "Now we only need p, which is the\r\n" );
document.write( "distance from the vertex to the focus \r\n" );
document.write( "and also the distance from the\r\n" );
document.write( "vertex to the directrix. \r\n" );
document.write( "\r\n" );
document.write( "So the distance from V(3,4) to F(3,2)\r\n" );
document.write( "is 2 units, so p = 2.\r\n" );
document.write( "\r\n" );
document.write( "In the equation\r\n" );
document.write( "\r\n" );
document.write( " (x - 3)² = 4p(y - 4) \r\n" );
document.write( "\r\n" );
document.write( "we can put in 2 for p and have\r\n" );
document.write( "\r\n" );
document.write( " (x - 3)² = 4(2)(y - 4)\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( " (x - 3)² = 8(y - 4)\r\n" );
document.write( "\r\n" );
document.write( "That's the equation, but now we have to\r\n" );
document.write( "draw the graph. \r\n" );
document.write( "\r\n" );
document.write( "Draw a line from the focus thru\r\n" );
document.write( "the vertex to the directrix.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now draw a square on the left of that line.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now draw a square on the right of that line:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Finally sketch in the parabola going through the vertex\r\n" );
document.write( "and the corners or the two squares:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "That's the graph.\r\n" );
document.write( "\r\n" );
document.write( "----------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "2. Write the equation for the circle with center (3,-4) \r\n" );
document.write( " _ \r\n" );
document.write( " and radius Ö5\r\n" );
document.write( "\r\n" );
document.write( " (x - h)² + (y - k)² = r²\r\n" );
document.write( "\r\n" );
document.write( "is the equation of the circle with center (h,k) and radius r. So we\r\n" );
document.write( "substitute (h,k) = (3,-4) \r\n" );
document.write( " _\r\n" );
document.write( " (x - 3)² + (y - (-4) )² = (Ö5)²\r\n" );
document.write( "\r\n" );
document.write( " (x - 3)² + (y + 4)² = 5\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " \r\n" );
document.write( "------------------------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "4. Write equation for the ellipse with foci (9,0) and (-9,0) and endpoints on the verticle axis of (0,2) and (0,-2).\r\n" );
document.write( "\r\n" );
document.write( "The equation of such an ellipse is\r\n" );
document.write( "\r\n" );
document.write( " (x - h)² (y - k)²\r\n" );
document.write( " -------- + ---------- = 1\r\n" );
document.write( " b² a²\r\n" );
document.write( "\r\n" );
document.write( "where (h,k) is the center\r\n" );
document.write( "a = half the length of the major axis\r\n" );
document.write( "and b = half the minor axis\r\n" );
document.write( "and a² = b² + c² where c is the distance from\r\n" );
document.write( "Center to foCus. (You can remember that because\r\n" );
document.write( "the words \"Center\" and \"foCus\" both have the \r\n" );
document.write( "letter C, and \"vertex\" doesn't.)\r\n" );
document.write( "\r\n" );
document.write( "Lets plot the foci F(-9,0) and F(9,0) and the \r\n" );
document.write( "ends of the minor (vertical) axes (0,-2) and\r\n" );
document.write( "(0,2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " \r\n" );
document.write( "Now we know that the center is halfway between the\r\n" );
document.write( "foci, and the origin (0,0) is halfway between the\r\n" );
document.write( "two foci, so we know that (h,k) = (0,0).\r\n" );
document.write( "\r\n" );
document.write( "We know that c is the distance from the Center to\r\n" );
document.write( "a foCus, which is 9 units, so c = 9 \r\n" );
document.write( "\r\n" );
document.write( "We also know that b = half the minor (vertical in this \r\n" );
document.write( "case) axis. Since the minor axis is the line from (0,-2)\r\n" );
document.write( "to (0,2), and it is 4 units long, then half the\r\n" );
document.write( "minor axis is 2, and so b = 2, so we can fill those\r\n" );
document.write( "into:\r\n" );
document.write( "\r\n" );
document.write( " (x - h)² (y - k)²\r\n" );
document.write( " -------- + ---------- = 1\r\n" );
document.write( " b² a²\r\n" );
document.write( "\r\n" );
document.write( "and get\r\n" );
document.write( "\r\n" );
document.write( " (x - 0)² (y - 0)²\r\n" );
document.write( " -------- + ---------- = 1\r\n" );
document.write( " 2² a²\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( " x² y²\r\n" );
document.write( " --- + ---- = 1\r\n" );
document.write( " 4 a²\r\n" );
document.write( "\r\n" );
document.write( "So all we need is a, which is half\r\n" );
document.write( "the length of the major axis. We\r\n" );
document.write( "can calculate a from the relation\r\n" );
document.write( "\r\n" );
document.write( " a² = b² + c²\r\n" );
document.write( " a² = 2² + 9²\r\n" );
document.write( " a² = 4 + 81\r\n" );
document.write( " a² = 85\r\n" );
document.write( "\r\n" );
document.write( " __\r\n" );
document.write( " a = Ö85 = 9.2 approximately\r\n" );
document.write( "\r\n" );
document.write( "This puts the vertices at V(-9.2,0) and V(9.2,0) and they\r\n" );
document.write( "are very close to the focal points:\r\n" );
document.write( " \r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now that we know that a² = 85, the equation of the\r\n" );
document.write( "ellipse is:\r\n" );
document.write( "\r\n" );
document.write( " x² y²\r\n" );
document.write( " --- + ---- = 1\r\n" );
document.write( " 4 85\r\n" );
document.write( "\r\n" );
document.write( "and we can sketch in the ellipse: \r\n" );
document.write( "\r\n" );
document.write( " graph(800,266.7,-12,12,-4,4) )}}}\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( " \n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
