Conic Sections Understanding the graphs of conic sections is made easier if you first begin with the simplest form of a conic section. These would be the graphs that are centered at the origin. If we can understand how these simpler versions are drawn then the more complicated versions are just modified versions of our basic graphs. Basic s Circle Ellipse Hyperbola Hyperbola (Left/Right) (Up/Down) Equation x2 +y 2 =r 2 +-l --.-l a2 b2 a2 b2 a2 b2 vertex (0,0) vertex (0,0) vertex (0,0) vertex (0,0) Variables r = The radius of the a = The distance from the a = The distance from a = The distance from (Other than x circle. Distance center to the edge of the ellipse the origin to the the origin to the from center to any on the x axis (note: a is under vertices of the curve vertices of the curve or y, point on the circles the x term). (Major axis, note in the (Major axis, note in the edge. b = The distance from the formula the x term is formula the y term is center to the edge of the ellipse positive so this is the positive so this is the on the y axis (note: b is under major axis). major axis). the y term). b = The distance from b The distance from the origin to the outer the origin to the outer edge of the minor axis edge of the minor axis (Note: the minor axis (Note: the minor axis does not have contain does not contain the the curves of the curves of the hyperbola) hyperbola) a&b: To generate the a&b: To generate the asymptotes of the asymptotes of the hyperbola look at hyperbola look at y=±x / ::rtat1 x2+y 2=4 7 2 2 Z+z 1 X2 Y2 4 25 jj 36 4
-3 Parabola (Up/Down) Parabola (Left/Right) Equation y = x = vertex (0,0) vertex (0,0) Foci, 1 1 2.. y=x_ x=y Directerix p is the distance from the vertex to the focus or the p is the distance from the vertex to the focus directerix (note: The focus and the point on the directrix or the directerix (note: The focus and the closest to the vertex are the same distance away from the point on the directrix closest to the vertex are vertex but in opposite directions). The directrix is the the same distance away from the vertex but horizontal line passing through the point p units below the in opposite directions). The directrix is the vertex (if up) or the point p units above (if down). In the vertical line passing through the point p units basic graph: 1 1 to the left of the vertex (if right) or the point 1 so J3 = p units to the right of the vertex (if left). 4p 4 (A rep resentativ / e example) / Let s look at some examples: x2 2 1. +-----=1 16 9 From the chart let s look at what we know about this graph This is the graph of an ellipse a2 = 16, b2 = 9; This means the distance from vertex to the edge of the ellipse on the major axis is a = 4 and the distance from the vertex to the edge of the minor axis is b = 3. c2 = 16 9 7, This means c = x/ axis and we know the foci lie on the major axis. The foci are c = fl units from the vertex on the major axis. 5 * Represent the foci 1 and since a is larger than b this means x is the major 4 lil I I I I P i I I -5 - -2-1 1 2 3 A 5-1 x -2-.4- -5
2)2 3)2 = What if we want to move the vertex so that the conic section is no longer centered at the origin? We can do this by using translation, let s look at an example: Example 2 2 2 2 2 (x-2) (y-3) ()..... + = 1 (This is still an ellipse based on its overall form + 1) 16 9 16 9 By subtracting 2 from x we now change the location of our x-values, the same thing happens with our y-values because of the subtraction by 3. > But since we are changing all of our x-values in the same way (every value gets subtracted by 2) then they all shift from where they were in the same direction and with the same number of spaces. > To determine what direction they all shifted in; ask yourself what value of x would make the x2) () 2 part 0? Notice in our first example the x2 and y2 terms are zero if both x & y are zero, hence our vertex is (0,0). For this example if x 2 then (x is zero. To find the shift for the y values do the same thing; you ll notice y 3 would make (y zero. This means our vertex is now at (2,3). > In general, if you see x-a then this is a shift to the right and if you see x+a then this will be a shift to the left. (a is positive) > In general, if you see y-a then this is a shift up and if you see y+a then this will be a shift down. (a is positive) > To generate the picture around this point we use the same principles that we used in the previous example for all the other values. Note: The other values were in terms of distance from the vertex, this means that once we know where the vertex is we can detennine the location of the other parts of the graph by using the distances measured from the new vertex. So now let s generate a graph o This is the graph of an ellipse o The vertex is at (2.3) o a2 =16, b2 =9;Thismeansa4andb=3 o = 169 = 7, This means c = f5 and since a is larger than b this means x is the major axis 6- Minr Axis j / -2 1 x 4 6 Let s summarize the translation idea in a chart.
Circle a2 Ellipse =r Equation (x-x +(yy 2 (xx =1 Variables than x or (Other vertex (x0, y0) vertex (x0, y0) r = The radius of the circle. Distance from center to any point on the circles edge. a = The distance from the center to the edge of the ellipse on the x axis (note: a is under the x term). b = The distance from the center to the edge of the ellipse on the y axis (note: b is under the y term). Foci, None Directerix a2 = (if a is larger than b, x is the major axis) or c2 (A representative example) j 7/ (if b is larger than a, y is the major axis) *c is the distance from the center of the ellipse to its foci. Note: The foci are always located on the major axis, the major axis is the longest axis. /! Parabola (Up/Down) Parabola (Left/Right) Equation 1 1 yy 0=(xx 0=(yy 4p 4p vertex (x0, y0) vertex (x0,y0) 0) xx 2 Foci, 1 2 x. x0 = (x x0) y Directerix 4-p p is the distance from the vertex to the focus p is the distance from the vertex to the focus or the directerix or the directerix (note: The focus and the (note: The focus and the point on the directrix closest to the point on the directrix closest to the vertex are vertex are the same distance away from the vertex but in the same distance away from the vertex but in opposite directions). The directrix is the horizontal line opposite directions). The directrix is the passing through the point p units below the vertex (if up) or the vertical line passing through the point p units point p units above (if down). to the left of the vertex (if right) or the point p units to the right of the vertex (if left). (A 1 representative / example) \,/ 7
J.4) a2 Equation I (xx Hyperbola (Left/Right) a2 b2 1 r (yy Hyperbola (Up/Down) 1 (xx b2 vertex (x0 y0) vertex (x0, y0) Variables a = The distance from the origin to the vertices of the curve a The distance from the origin to the vertices (Other (Major axis, note in the formula the x term is positive so this is of the curve (Major axis, note in the formula the major axis). the y term is positive so this is the major axis). lban x or b = The distance from the origin to the outer edge of the minor b = The distance from the origin to the outer 3 ) axis (Note: the minor axis does not have contain the curves of edge of the minor axis (Note: the minor axis the hyperbola) does not contain the curves of the hyperbola) a&b: To generate the asymptotes of the hyperbola look at a&b: To generate the asymptotes of the. hyperbola look at xx 0) 0 = ±(yy 0) OC1, a + b = c c is the distance from the vertex to the foci, the a + b = c Directerix foci lie on the major axis c is the distance from the vertex to the foci, the foci lie on the major axis (A // /. 22 ±(xx representative / N, 2 5 / example) 7, / 2 6 Example 3 (x1) 25 9 2,4)2 ( 2 ( 2 25 9 > This is a hyperbola (left/right) because it is of the form > We know the vertex is shifted to (1,4) 2 =25+9 =34 = 1 with the x term leading. > Weknowa=5 andb3, so c=a2 ±b > We know the coordinates of the foci are along the major axis which is the line y 4 (the horizontal line that the left and right vertices lie on). This means they are c units to the left and right of the x coordinate of the vertex so (1 (6.83,4) > We know that y 4 = ± (x 1) for the asymptotes. and (1 + /,4) which is approximately (-4.83.4) and (See next page for sketch)
If we sketch this, it will look like 1 4-4