t’s start with a little problem… e fact that k = 2p is twice the focal length and hal width to determine a Cartesian equation of the par polar equation is given. 12 3 3cos r 4 1 cos 1, 4 e k The graph??? 2 k p , so… 2,4 8 p p Vertex: , 2,0 hk And since the parabola opens left, the equation is: 2 8 2 y x
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Let’s start with a little problem… Use the fact that k = 2p is twice the focal length and half the focal width to determine a Cartesian equation of the.
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Let’s start with a little problem…Use the fact that k = 2p is twice the focal length and half thefocal width to determine a Cartesian equation of the parabolawhose polar equation is given.
12
3 3cosr
4
1 cos
1, 4e k
The graph???
2k p , so… 2,4 8p p
Vertex: , 2,0h k And since the parabola opens left, the equation is:
The midpoint M of the line segment PQ with endpointsP(x , y , z ) and Q(x , y , z ) is1 1 1 2 2 2
A Quick ExampleA Quick Example
, 2 17d P Q
Find the distance between the points P(–2, 3, 1)and Q(4, –1, 5), and find the midpoint of the linesegment PQ.
1,1,3M Can we verify these answers with a graph?Can we verify these answers with a graph?
Planes and Other Planes and Other SurfacesSurfaces
0Ax By C
We have already learned that every line in the Cartesianplane can be written as a first-degree (linear) equation in twovariables; every line can be written as
How about every first-degree equation in three variables???
They all represent planes in Cartesian space!!!
Planes and Other Planes and Other SurfacesSurfaces
0Ax By Cz D
Equation for a Plane in Cartesian SpaceEquation for a Plane in Cartesian Space
where A, B, and C are not all zero. Conversely, everyfirst-degree equation in three variables represents aplane in Cartesian space.
Every plane can be written as
Guided PracticeGuided Practice12 15 20 60x y z Sketch the graph of
Because this is a first-degree equation, its graph is a plane!
Three points determine a plane to find them:
15 4 3
x y z Divide both sides by 60:
5,0,0It’s now easy to see that the following points are on the plane:
0,4,0 0,0,3Now where’s the graph???Now where’s the graph???
Guided PracticeGuided PracticeSketch a graph of the given equation. Label all intercepts.