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So the equation is:

$$ x^2 + 2y^2 - 6x + 4y + 7 = 0 $$

Find the coordinates of the center, the foci, and the vertex or vertices.

What I did was put the equation in the form: $$ \frac{(x-3)^2}{4}+ \frac{(y+1)^2}{4} = 1 $$

Now based on that, I said the center is at $(3,-1)$, the foci is at ~+- 2.45 (since $c = \sqrt {a^2 + b^2}$ ). so the coordinates of that are $(3+2.45,-1)$ and $(3-2.45,-1)$ and the vertices are $(1,-1)$ and $(5,-1)$. I also went ahead and found the asymptote, which is just done by setting the equation to $0$, correct?

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What asymptote? This is an ellipse. And I get a different "standard form." – André Nicolas Jun 10 '12 at 20:11
@AndréNicolas perhaps the OP means the directrix? – Shaktal Jun 10 '12 at 20:13
Why do you divide both $x^2$ and $y^2$ by four in your transformed equation? – krlmlr Jun 10 '12 at 20:14
For future reference, you may be able to get quicker answers from Wolfram Alpha – Daniel Littlewood Jun 10 '12 at 20:15
Note the spelling of the plural of "vertex". It is "vertices", just as the plural of "matrix" is "matrices", of "axis" is "axes", of "basis" is "bases", of "thesis" is "theses", etc. (I corrected it in several places in the posting.) – Michael Hardy Jun 10 '12 at 20:29
up vote 2 down vote accepted

The equation should be $$\frac{(x-3)^2}{4}+\frac{(y+1)^2}{2}=1.$$ You've correctly identified the center and vertices. The focal length should be $\sqrt{a^2-b^2}$, not $\sqrt{a^2+b^2}$. Ellipses don't have asymptotes, you're thinking of hyperbolae.

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Is the focal axis parallel to the x-axis? – Nick Jun 10 '12 at 20:52
Since $4>2$, yes. If it had been (for example) $\dfrac{(x-3)^2}{2}+\dfrac{(y+1)^2}{4}=1$, then the focal axis would have been parallel to the $y$-axis, instead. (Recall that the focal axis is the major axis.) – Cameron Buie Jun 10 '12 at 20:55
Hmm, howcome you said the verticies have to change? Arnt they at (1,-1) and (5,-1)? – Nick Jun 10 '12 at 20:59
Ah! Right you are. I'll fix it. (Bonehead mistake. Forgot to take the square root.) – Cameron Buie Jun 10 '12 at 21:04
Awesome. Yes I made a typo in my original question, I did get the same solution as you. But I got the foci wrong because I did + instead of -... thanks alot! – Nick Jun 10 '12 at 21:08

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