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$\frac{(x+1)^2}{16} + \frac{(y-2)^2}{9} = 1.$

I just started conics, but I thought you would multiply both sides by $16$ and then $9$ and then expand, which would get you $x^2 +y^2+2x-4y+5$.

Both signs are the same, but the foci is supposed to be: $(−1 \pm \sqrt{7},2)$, which I don't get.

I thought it was supposed to be a circle. both squared variables are multiplied by the same number and have the same signs.

I tried to identify it off of:

Are both variables squared?      No: It's a parabola.     Yes: Go to the next test....     Do the squared terms have opposite signs?          Yes: It's an hyperbola.         No: Go to the next test....         Are the squared terms multiplied by the same number?              Yes: It's a circle.             No: It's an ellipse. 

Gotten from purplemath: http://www.purplemath.com/modules/conics.htm

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    As an aside, your link greatly overstates the 'difficulty' of slanted conics.2012-06-29

2 Answers 2

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If you take $\frac{(x+1)^2}{16} + \frac{(y-2)^2}{9} = 1$ and multiply both sides by $16$ and then $9$ you get $9(x+1)^2 + 16(y-2)^2 = 144.$ If you expand you get $9x^2+18x+16y^2-64y-71 =0$ which is not what you had.

In fact your equation is of a ellipse centred at $(-1,2)$ with a horizontal semi-major axis of $ \sqrt{16} =4$ and a vertical semi-minor axis of $ \sqrt{9} = 3$.

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    Thanks!!!!!!!!!!! I was like how is that different than a regular fraction, but 9 goes to one fraction, and the 16 goes to the other. had a brain lapse and treated everything like it was getting cancelled out.2012-06-29
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It's not a hyperbola. It's an ellipse.

In fact, here it is:

enter image description here

Let's go a bit further and see what we can say about it, since you just started conics and all. It's equation is $\dfrac{(x+1)^2}{16} + \dfrac{(y-2)^2}{9} = 1$.

The standard form is $\dfrac{x^2}{r_x^2} + \dfrac{y^2}{r_y^2} = 1$, where $r_x$ is the length of the $x$-radius and $r_y$ is the length of the $y$-radius. If $r_x > r_y$, then $r_x$ is called the length of the major axis and $r_y$ is called the length of the minor axis. The $+1$ with the $x$ moves the center to the left by $1$,and the $-2$ with the $y$ moves the center up by $2$. That's why the center is at $(-1,2)$ in the graph above.

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    i meant the coeffient of the x and y squared. I was going off of this: Are both variables squared? No: It's a parabola. Yes: Go to the next test.... Do the squared terms have opposite signs? Yes: It's an hyperbola. No: Go to the next test.... Are the squared terms multiplied by the same number? Yes: It's a circle. No: It's an ellipse. gotten from purplemath: http://www.purplemath.com/modules/conics.htm2012-06-29