Chapter 10: Analytic Geometry


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1 10.1 Parabolas Chapter 10: Analytic Geometry We ve looked at parabolas before when talking about the graphs of quadratic functions. In this section, parabolas are discussed from a geometrical viewpoint. All parabolas in this section have their vertex at the origin. A parabola is the set of points in the plane equidistant from a fixed point F called the focus and a fixed line l called the directrix. The vertex V of a parabola lies halfway between the focus and the directrix. The axis of symmetry is the line that runs throught the focus perpendicular to the directrix. The focal diameter of a parabola is the length of the line segment that runs through the focus perpendicular to the axis with endpoints on the parabola. Parabola with Vertical Axis Parabola with Horizontal Axis Equation: x 2 = 4py y 2 = 4px Vertex: (0, 0) (0, 0) Focus: (0, p) (p, 0) Directrix: y = p x = p Focal diameter: 4p 4p Opens: upward if p > 0 to the right if p > 0 downward if p < 0 to the left if p < 0 A way to distinguish between these two types is that the axis of symmetry corresponds to the variable that is not squared. (Why? If you have the equation x 2 = 4py and you replace x by x, the equation does not change, which means this equation has symmetry with respect to the yaxis.) Once you know the axis of symmetry, you know that the focus must lie on this axis and that the directrix must cross through this axis. 1
2 Examples: 1. Find the focus, directrix, and focal diameter of the parabola x 2 = 5y. Then sketch a graph. 2. Find an equation for the parabola that has its vertex at the origin with directrix y = Find the focus, directrix, and focal diameter of the parabola 5x + 3y 2 = 0. Then sketch the graph. 4. Find an equation of the parabola that has its vertex at the origin and has focus (5, 0). 2
3 10.2 Ellipses An ellipse is the set of all points where the sum of the distances from two fixed points F 1 and F 2 is a constant. These two fixed points are called the foci (plural of focus) of the ellipse. The minor axis is the shorter axis of the ellipse. The major axis is the longer axis of the ellipse. The vertices of an ellipse are where the ellipse touches the major axis. For this section, the center of the ellipse will always be the origin. Ellipse with Horizontal Major Axis Ellipse with Vertical Major Axis Equation: x 2 a 2 + y2 b 2 = 1, a > b > 0 x 2 b 2 + y2 a 2 = 1, a > b > 0 Vertices: (±a, 0) (0, ±a) Major Axis: Horizontal, length 2a Vertical, length 2a Minor Axis: Vertical, length 2b Horizontal, length 2b Foci: (±c, 0) (0, ±c) c 2 = a 2 b 2 c 2 = a 2 b 2 The way to distinguish between these two types is that the larger number a always appears in the denominator below the variable that corresponds with the major axis. (The xvariable corresponds with a horizontal major axis and the yvariable corresponds with a vertical major axis.) Once you know which is the major axis, you know that the vertices and foci must lie on this axis. The eccentricity e of an ellipse is a measure of how elongated the ellipse is. e = c a. The eccentricity of an ellipse always satisfies 0 < e < 1. The closer e is to 0 the more circular the ellipse is. The closer e is to 1, the more elongated or stretched the ellipse is. 3
4 Examples 1. Find the vertices, foci, the lengths of the major and minor axes, and the eccentricity of the ellipse = 1. Then sketch the graph. x y Find an equation for the ellipse with foci (±5, 0) and length of major axis Find the vertices, foci, the lengths of the major and minor axes, and the eccentricity of the ellipse 4x 2 + y 2 = 16. Then sketch the graph. 4. Find an equation for the ellipse with foci (0, ±3) and vertices (0, ±5). 4
5 10.3 Hyperbolas A hyperbola is the set of all points where the difference of the distances from two fixed points F 1 and F 2 is a constant. These two fixed points are called the foci of the hyperbola. A hyperbola consists of two branches. The segment joining the two branches of the hyperbola is called the transverse axis, and the vertices are where the hyperbola touches the transverse axis. For this section, the center of the hyperbola will always be the origin. The asymptotes of a hyperbola are the lines which the branches are approaching. Hyperbola with Horizontal Transverse Axis Hyperbola with Vertical Transverse Axis Equation: x 2 a 2 y2 b 2 = 1, a > 0, b > 0 y 2 a 2 x2 = 1, a > 0, b > 0 b2 Vertices: (±a, 0) (0, ±a) Transverse Axis: Horizontal, length 2a Vertical, length 2a Foci: (±c, 0) (0, ±c) c 2 = a 2 + b 2 c 2 = a 2 + b 2 Asymptotes: y = ± b a x y = ± a b x The way you can tell the difference between these two types is that the transverse axis corresponds to the variable which is positive in the equation (doesn t have the negative sign in front of it). With hyperbolas, we don t care which variable has the larger number under it. Again, once you know which is the transverse axis, you know that the vertices and foci lie on this axis. Note: In both cases, the slope of the asymptotes is ± the square root of the number under y 2 divided by the square root of the number under x 2. (You can remember this by thinking that the slope of a line is change in y over change in x. ) 5
6 Examples 1. Find the vertices, foci, and asymptotes of the hyperbola x2 4 y2 9 = 1. Then sketch the graph. 2. Find an equation for the hyperbola with foci (±5, 0) and vertices (±3, 0). 3. Find the vertices, foci, and asymptotes of the hyperbola y2 9 x2 16 = 1. Then sketch the graph. 4. Find an equation for the hyperbola with vertices (0, ±6) and asymptotes y = ± 1 3 x. 6
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