Question

find the center, vertices, foci, eccentricity, length of major axis, and length of minor axis for (x-5)^2/4+(y+3)^2/16=1

Answer 1

Do you know what conic it is?

Answer 2

sort of its like hyperbolas, parabolas, ellipses, and i believe circles

Answer 3

Can you tell which one from your text book?

Answer 4

it is all of them so far and i have to draw the graph as well :/

Answer 5

Each conic has a characteristic and a standard form.
If you study conics, the first thing you need to know is to tell which one it is!

For example,
Equation of every circle can be reduced to:
\(\large (x-a)^2+(y-b)^2=r^2\)

Answer 6

Does it look like a circle?

Answer 7

I think its an ellipse since someone told me that this is what they usually look like if i remember correctly

Answer 8

Yes, it is an ellipse, with the standard equation:

Answer 9

\[\frac{ (x-5)^2 }{ 4 }+\frac{ (y+3)^2 }{ 16 }=1\]

Answer 10

\(\large \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=-1\)

Answer 11

Oops:
\(\large \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)

Answer 12

where (h,k) is the centre of the ellipse,
a and be are the axes.
If a>b, the major (longer) axis is horizontal, and
if a<b, the major axis is vertical.

Answer 13

ok so far?

Answer 14

a and b are the distances from the centre to the vertices.

Answer 15

Can you find h,k, a,b from the given equation, in comparison with the general equation?

Answer 16

h=5 and k=-3 i think and the a=2 and b=4

Answer 17

am i right?

Answer 18

That is correct!

Answer 19

So a is x-distance between centre and vertex, b is y-distance of same.

Answer 20

Do you know the definition of major and minor axes?

Answer 21

*definitions

Answer 22

major is the square root of the lower number times 2 and major is the squareroot of the highest number times 2?

Answer 23

exactly, in other words, major axis is 2a or 2b, whichever is greater.
Similarly for minor.

Answer 24

How about eccentricity?

Answer 25

that i can't remember :/

Answer 26

eccentricity = sqrt(1-(minor axis/major axis)^2)

Answer 27

\[\sqrt{1-4/8}?\]

Answer 28

(4/8)squared!

Answer 29

so -1-1/2? i am sorry i dont get this part

Answer 30

http://en.wikipedia.org/wiki/Eccentricity_(mathematics)
in case you need a reference.

eccentricity = sqrt(1-(4/8)^2)=sqrt(1-1/4)=sqrt(3/4)=sqrt(3)/2

Answer 31

sorry, afk.

Answer 32

oh ok

Answer 33

what about the vertices and foci?

Answer 34

Vertices: (h \(\pm a, k \pm b)\)
c=sqrt(major^2-minor^2)
foci:
\((h \pm c, k)\) for a>b
\((h, k \pm c)\) for a<b