Wednesday, June 3, 2009

Conics

Parabolas


Definition: the set of all points (x,y) in a plane that are equidistant from a fixed line, also known as the directrix, and a fixed point, the focus, which is not on the line.

Vertex - the midpoint of the directrix and focus of the parabola

Axis - the line perpindicular to the directrix that passes through the vertex


The graph to the below is a parabola with the vertex at the origin with the equation:



That equation is simply the standard equation
when h and k are set to zero







If you switch x and y, you end up with a graph looking like this:

The Standard Equation of a Parabola:

(opens up/down)

(opens left/right)

p= distance from vertex to focus
focal width: 4p




*How to find "P" if you have a graph:

Find a point on the graph (not the origin) and sub in to the standard equation

Check to make sure you use the correct standard equation, based on the direction the graph opens

Solve for P

HINT:
the parabola always opens towards the non-squared term

Ellipses

Definition: An ellipse is the set of all points in a plane, the SUM of whose distances from two foci is constant

Vertices: the two points which the line through the foci intersect


Standard Equation of an Ellipse
When the ellipse has a longer horizontal axis:







When the ellipse has a longer vertical axis:



Finding the Foci:





Eccentricity:
measures the ovalness of an ellipse



When e is closer to zero, the ellipse is more circular
When e is closer to one, the ellipse is more elliptical



Hyperbolas
Definition: A hyperbola is the set of all points in a plane, the DIFFERENCE of whose distances from two foci is constant

In the hyperbola, there are two vertices instead of one, and the two curved lines open in opposite directions



The STANDARD EQUATION of a hyperbola is:
(when the hyperbola "branches" open left/right)



(when the hyperbola "branches" open up/down)





(slant asymptote)


Eccentricity: When e is larger, the branches are flatter

When e is smaller, (closer to 1) the branches are more pointed



Circles
Definition: A circle is the locus of all points equidistant from a point

Standard equation:


where (h,k) is the center
and r is the radius










PRACTICE PROBLEMS:

1. Write the equation of the locus of points in a plane tangent to x=3 and equidistant from the point (1, -4)


2. Determine the coordinates of the foci for





3. Write the equation of the hyperbola whose foci are (-5, -2) and (-5, 10) and whose vertices are(-5, 0) and (-5, 8)


A point of the parabola is (6, -1). Find:


a) P
b) vertex
c) focus
d) equation of directrix



5. If the eccentricity of
is 3/4, find b.




6. Find the coordinates of the center and radius of a circle whose equation is





7. Write the equation of the locus of points in a plane equidistant from x = 8 and (2, 4)





8. Find the coordinates of the vertices of the hyperbola







9. Find the eccentricity of an ellipse with c = 4.5 and a = 5.
Does it closely resemble a circle?



10. Write the equation of a hyperbola with vertices (0, ±3)
and asymptotes of y = ±3x












ANSWERS:





2. (-5, 2) and (-5, 6)





4. a) 2; b) (2, -3); c) (2, -1); d) y= -5



6. center: (1, -3)





8. vertices: (2, 2) (2, -10)


9. 9/10; No

14 comments:

Nicole S. said...

sorry the practice problems and answers are messed up, we had a lot of problems with those but we are working on it =)

Nicole S. said...

Hey everyone.. so for some reason a lot of the equations aren't in the right places.. like they are before the text if you know what i mean.. i tried to fix it but it wouldn't work =(. Maybe it's just on my computer? I don't know.. sorry about that!!!

Kate Nowak said...

Nicole I like the way you have it laid out. I think it's fine. But you can drag images around in "compose" mode and put them where you want. You can also add lines spaces at the beginning of text to move the text down.

Kate Nowak said...

You have your hyperbola equations backwards. When the y term comes first, it opens up/down.

Nicole S. said...

oh jeeez ok i fixed it =)

Steve said...

I like the hyperbola diagram. It's really clear and easy to understand.

kelseyyyy said...

I agree with steve! I never really understood hyberbolas until now because it was so easy to understand!

Ellen + Kelsey said...

Hey guys just wanted to say that your post is really helpful and clear and i know no one will probably get to answer this but just for clarification, in an ellipse the larger a value or b value is in the denominator of the first fraction? So then if you look at just the equation of an ellipse how do you know if the larger a or b value refers to the width of the ellipse or the height of the ellipse?

Ellen + Kelsey said...

oh i got it nevermind!

Jon R. said...

Great post! Really makes a hard unit with lots of things to memorize a lot easier. One question though if by chance you are able to answer it before tomorrow. If the SA of a hyperbola is the square root of (b^2/a^2)then doesn't that come out to be just b/a?

Steve said...

I'm pretty sure the SA of a hyperbola is the (y junk) = (thing under y / thing under x) (x junk).

For example, (y-3)^2 (x+2)^2
-------- - -------- = 1
16 4

The SA would be (y-3) = + or - (4/2) (x + 2).

Steve said...

Formatting got messed up, the (y-3) should be over the 16 and the (x+2) should be over the 4.

Steve said...

Sorry for the multiple posting (can't figure out how to edit), but I just checked out their graph closer. The only reason b/a works (which is the same as rad((b^2/a^2)) is because the origin for that hyperbola is at (0,0).

Kate Nowak said...

The slope is always going to be y-distance/x-distance (as this is written: either b/a or a/b, depending on which is under x and which is under y).

You can always use the center (h,k) to write the equation of the line in point-slope form: y-k=(b/a)(x-k).

If you insist on sticking with slope-intercept form (y=mx+b), you are going to have to find the 2 different y-intercepts by your favorite method.