Question

Identify the type, center, and foci of the conic section. 11x^2 – 2y^2 + 66x – 16y + 45 = 0 (I'll post what I work I have done so far)

Answer 1

My work:

\[11x ^{2}+66x-2y ^{2}-16y=-45\]\[11(x ^{2}+6x)-2(y ^{2}+8y)=-45\]\[11(x ^{2}+6x+(3^{2}))-2(y ^{2}+8y+(4^{2}))=-45+11(3^{2})+(-2)(4^{2})\]\[11(x ^{2}+6x+9)-2(y ^{2}+8y+16)=22\]\[11(x+3)^{2}-2(y+4)^{2}=22\]\[\frac{ 11(x+3)^{2} }{ 22 }-\frac{ (y+4)^{2} }{ 22 }=1\]\[\frac{ (x+3)^{2} }{ 11 }-\frac{ (y+4)^2 }{ 2 }=1\]
I'm not sure how to tell what type of conic section it is. Can someone help me? Textbooks are not very helpful.

Answer 2

@whpalmer4 Do you know how to do this? If so, do you think you could help me?

Answer 3

@thomaster Are you good at these types of problems?

Answer 4

@Mertsj Can you help me?

Answer 5

@rosedewittbukater when the sign is not the same you have a hyperbola. Because the \(x\)-term is positive, we have a *horizontal* hyperbola.

Answer 6

@oldrin.bataku how do you know its a hyparabola?

Answer 7

The negative sign between the two terms on the left tells you. If it were a plus sign, it would be an ellipse.

Answer 8

And the first denominator should be 2 not 11

Answer 9

Oh yeah, I wrote that wrong. That's what I meant to put in the denominator.

Answer 10

And the denominator of the y term should be 11

Answer 11

Umm not really that's confusing. @oldrin.bataku

Answer 12

So since it's a hyperbola, how do I find the center and foci?

Answer 13

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

Answer 14

That is the equation of a hyperbola whose center is (h,k). Make your equation look like that and figure out what (h,k) is. That is your center.

Answer 15

Then find c. Use:
\[a^2+b^2=c^2\]

Answer 16

So the center is (-3, -4).

Answer 17

The foci are c units to the right and c units to the left of the center.
And yes, your center is correct.

Answer 18

And the foci is (+/-12.85, 0)?

Answer 19

Wait I did the math wrong...

Answer 20

\[c^2=11+2=13\]

Answer 21

\[c=\sqrt{13}\]

Answer 22

Foci:
\[(-3\pm \sqrt{13},-4)\]

Answer 23

Oh ok thank you so much!!! I really appreciate it. :)

Answer 24

yw