10.5 Planes Lines Notes

10.5 Planes Lines Notes - 
 1 0 .5 

L in e s 
a n...

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 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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