10.5 Planes Lines Notes

# 10.5 Planes Lines Notes -   1 0.5   L in e s  a n d...

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Unformatted text preview:   1 0 .5   L in e s  a n d  P l a n e s   Conics  I. Parabolas  Equation     Axis     Opens  x 2 = 4 py     y‐axis          up  x 2 = !4 py     y‐axis     down  y 2 = 4 px     x‐axis      right  y 2 = !4 px     x‐axis          left      x 2 = −4 y and y 2 = 6 x .  Below are graphs of    -3 -2 -1 1 2 3 0.6 - 0.5 0.4 0.2 - 1.0 -0.2 - 1.5 0.5 1.0 1.5 2.0 2.5 3.0 -0.4 - 2.0            -0.6         II. Ellipses  x 2 y2 + 2 =1 2 b Ellipses have the equation   a  where the center of  the ellipse is the origin.  The ellipse extends “a” units to the left  and right of the origin and “b” units above and below the  x 2 y2 + =1 9 origin.  The example below is of  4 .    3 2 1 -2 -1 1 2 -1 -2 -3           III. Hyperbolas  x 2 y2 y2 x 2 ! 2 =1 ! 2 =1 2 2 b b Hyperbolas are of the form   a  or  a .  The  hyperbola crosses the axis of the positive variable.  The  y2 x 2 y2 2 ! x = 1! and ! ! =1 4 9 examples below are of   16 .    10 5 -3 2 - 123 4 -5 2 -4 -2 2 4 -2 - 10 -4                   3 – space  z y   x   Ex.   R2:  x = 2  y x   Ex.  R3:   x = 2  z y   x   Equation of a plane:                Ax + By + Cz = D  Graph by finding the x, y, and z intercepts.  Ex. 2x + y – z = 4  z y             x   Ex. 2x + y = 4  z y               x Equation of a plane:   We need a point  P(a,b,c) and a normal vector  ! ! ! ! n = n1i + n2 j + n3 k   n1 x + n2 y + n3 z = n1 , n2 , n3 ! a, b, c   Ex.  Find the equation of the plane through P(4, 0, 6) normal to  ! n = !1, 7, 3 .            Equation of a line:  We need a point P(a,b,c) and a parallel vector  ! v = v1 , v2 , v3   x = a + v1t y = b + v2t z = c + v3t     Ex.  Find the equation of the line through points P(4,0,6) and  Q(1, 1, ‐1)                        Ex.  Find the equation of the plane through the origin which  contains the line   x = 2+t y = !1 z = !2t                   Do: 1. Find the equation of the line through P(‐4, 0, 5)  perpendicular to the plane 2y = 5.    2.  Find the equation of the plane through the point   P(‐4, ‐4, 5) and parallel to the lines  x = 2 + t !!!!!!!!!!!!!!!!!!!!!!! x = 4 ! 3s y = !3!!!!!!!!!!!!!!!!!!!!!!!!!!! y = s            z = 3t !!!!!!!!!!!!!!!!!!!!!!!!!!!! z = 6 ! 7 s     ...
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