## Need Help Ellipse and Hyperbola Equations

Trigonometric ratios and functions, the unit circle, inverse trig functions, identities, trig graphs, etc.
agenta1013
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### Need Help Ellipse and Hyperbola Equations

I have two conic equations that I've been unable to put in standard form without getting at least one fraction, which messes up the rest of the steps.

The first: $2x^2-6x+3y^2+12y-6=0$

I know this is an ellipse because the $x^2$ and $y^2$ terms have the same sign.

I need to put it in the standard form $(x-h)^2/a^2+(y-k)^2/b^2$ = 1 by factoring out terms and completing the square:

$2(x^2-3x)+3(y^2+4y)=6$

Completing the square causes this issue with the fraction in the x term:

Halving -3 and squaring it gets the fraction $9/4$, so I get:

$2(x^2-3x+(9/4))+3(y^2+4y+4)=6+2*(9/4)+3*4$

The right side simplifies to:

$2(x^2-3x+(9/4))+3(y^2+4y+4)= (45/2)$

The y expression is an easy perfect square, so I get:

$2(x^2-3x+(9/4))+3(y+2)^2 = (45/2)$

I can't get the x expression into a perfect square, and I've tried multiple approaches such as:
$(1/4)(2x-3)^2+3(y+2)^2 = (45/2)$

Dividing both sides by $(45/2)$ yields even more of a fraction mess, and I still can't get the numerators $(x-h)^2$ and $(y-k)^2$ isolated.

The second: $2x^2-6x-3y^2+12y-6=0$ --a hyperbola

After putting these in standard form, I need to find the center, vertices and foci--but I obviously can't do this until they're in standard form. Sorry about the length of this post, but I wanted to add everything I've tried so far. I'm about at my wit's end with these, so any help would be awesome!

Thanks

nona.m.nona
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### Re: Need Help Ellipse and Hyperbola Equations

Kindly please clarify how fractions are preventing a solution. Fractions, in and of themselves, are perfectly valid within the contexts of conic equations.

little_dragon
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### Re: Need Help Ellipse and Hyperbola Equations

The first: $2x^2-6x+3y^2+12y-6=0$
2x^2 - 6x + 3y^2 + 12y = 6
2(x^2 - 3x) + 3(y^2 + 2y) = 6
2(x^2 - 3x + 9/4) + 3(y^2 + 2y + 1) = 6 + 2(9/4) + 3(1)
2(x - 3/2)^2 + 3(y + 1)^2 = 6 + 9/2 + 3
2(x - 3/2)^2 + 3(y + 1)^2 = 27/2
2(2/27)(x - 3/2)^2 + 3(2/27)(y + 1)^2 = 1
(4/27)(x - 3/2)^2 + (2/9)(y + 1)^2 = 1
(x - 3/2)^2 / (27/4) + (y + 1)^2 / (9/2) = 1

so center (h,k) = (3/2, -1)
27/4=6.75
9/2=4.5
so a^2=27/4, b^2=9/2
a=3sqrt(3)/2, b=3/sqrt(2)

its messy but it works

agenta1013
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### Re: Need Help Ellipse and Hyperbola Equations

Thank you!

This is confusing with fractions because there is no example in my textbook and the professor did not go over any examples of these with fractions either, only with simple whole numbers.

buddy
Posts: 197
Joined: Sun Feb 22, 2009 10:05 pm
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### Re: Need Help Ellipse and Hyperbola Equations

& now you see why: stuff w/ fractions is messy!