(3,4) focus. y=1 directrix
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1 Math : Conic Sections Parabolas, Ellipses, Hyperbolas Parabolas: Definition: A parabola is the set of all points in a plane such that its distance from a fixed point F (called the focus) is equal to its distance from a fixed line d (called the directrix) Example: 8( y ) ( x 3) (3,4) focus y=1 directrix Let s try and derive the formula of a parabola directly: For simplicity assume the vertex is the origin. And its directrix is parallel to the x- axis. If the focus is at (0, p), then the equation of the directrix must be y = -p. Let P(x,y) be a point on the parabola: (0, p) (x,y) (x,-p)
2 Form 1 opens vertically If vertex is (0,0) this form is The general form of a parabola: 4py x or y 1 4p x Focus: (0, p) Directrix: y = -p x=0 axis of symmetry p = focal distance (distance from vertex to focus (opens up if p > 0 down if p < 0)) If the vertex is (h, k) then everything is translated horizontally and vertically: 4p( y h k) ( x ) Focus (h, k+p) Directrix: y = k-p x=h is axis of symmetry You can do the same proof for a parabola that has a vertical line for the directrix and opens horizontally Form opens horizontally If vertex is (0,0), then this form reduces to : 4px y or x 1 4p y Focus: (p, 0) Directrix: x = -p y=0 axis of symmetry if p > 0, opens to the right if p < 0, opens to the left If the vertex is (h,k) then everything is translated horizontally or vertically 4p( x k h) ( y ) Focus: (h+p, k) Directrix: x = h-p y=k is axis of symmetry The focal width of a parabola: 4p
3 Ellipse: Definition: An ellipse is the set of all points in the plane, the sum of whose distance from two fixed points is a positive constant. For an ellipse consider the following drawing the center of this ellipse is the origin and the major axis length is a and the minor axis length is b, The distance from the center to a focus is c (-a, 0) (0, b) (-c,0) (c,0) (a, 0) (0, -b) The points (a, 0) and (-a, 0) are called vertices (endpoints of the major axis) and the points (0, b) and (0, -b) are endpoints of the minor axis. The big dots are the foci with coordinates (c, 0) and (-c, 0) We can use this drawing to find the general form of an ellipse with center (0, 0) and point (x, y) on the ellipse and foci (c, 0) and (-c, 0) Standard equations of an ellipse with center at the origin The graph of x y x y 1 or 1 with a>b>0 is an ellipse with center (0, 0). The length a b b a of the major axis is a, the length of the minor axis is b. The focal distance c from the origin, where c a b
4 If the center of the ellipse is (h,k), by a simple transformation the equations of an ellipse become ( x ) h ( y) k a b ( x ) h ( y) k 1 or 1 where again a > b > 0 b a Hyperbolas: A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F1 and F (the foci) is a constant For hyperbola that opens horizontally: (-c,0) (-a,0) (a,0) (c,0) x y 1, has foci ( c,0), vertices ( a,0) and asymptotes a b c a b y b x, Where a For the hyperbola that opens vertically: (0, a) (0, -a) y x 1, has foci (0, c), vertices (0, a) and asymptotes a b c a b y a x, Where b
5 Of course if the center is NOT (0, 0) these hyperbolas are translated horizontally and vertically Examples/Practice: 1) Sketch the graph of the following parabola: ( y 1) 4( 5) x, find the vertex, focus and equation of directrix ) Sketch the following ellipse and indicate the vertices, endpoints of the minor axis and foci 49x 5y 15 3) Find the vertices, foci and asymptotes of the hyperbola and sketch its graph y 6x y 36x 44 4) Find the vertex, focus and directrix of the following parabola x x y 3 0 5) Find the equation in standard form of the parabola with focus (, -3) and directrix x= 5 6) Find the equation of the hyperbola with vertices (-1,), (7, ) and foci (-,) and (8, ) 7) If an ellipse centered at (0,0) with major axis on the x-axis is rotated around the x-axis, what is the volume of the resulting solid?
6 10.6: Polar Equations of Conics You can define all the conic sections in terms of a focus and directrix, not just the parabola. This representation is useful especially in astronomy Theorem: Let F be a fixed point (called the focus) and let l be a fixed line (called the directrix) in a plane. Let e be a fixed positive number (called eccentricity). The set of all points P in the plane such that is a conic section, The conic is a. an ellipse if e < 1 b. a parabola if e = 1 c. a hyperbola if e > 1 Proof: You can solve the equation: r e( d r cos( )) for r to get ed r 1 ecos( ). It is not too hard to see the following (note in each case the focus is at the origin) Theorem: A polar equation of the form
7 ed ed r or r 1 ecos( ) 1 esin( ) represents a conic section with eccentricity e. If e < 1 then it is an ellipse, If e > 1 then it is a hyperbola and if e =1 then it is a parabola Example: Find the polar equation of each of the following conics a. parabola with directrix x = -4 b. Ellipse with eccentricity ½ and r 4sec as the directrix c. Hyperbola with eccentricity of and directrix of y = - Example: Find the eccentricity, identify the conic, give an equation of the directrix and sketch the conic for the following 1 a) r 3 10cos b) 8 r 4 3sin Note: In an ellipse if eccentricity is 0 the ellipse is a circle. As e approaches 1, the ellipse get more elongated. If e =1 we have a parabola and if e> 1 the further away from 1 the more steep the hyperbola will become.
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