Determine wether `2x^2+3y^2+4x-6y-13=0` is a parabola, ellipse, or hyperbola. Give the center, foci, directrix...etc
Please show how to differentiate between the conic sections in this problem and show how to solve.
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Let's rewrite equation so we can see what kind of conic it describes.
We can write this as
`2(x+1)^2+3(y-1)^2=18` now devide whole equation with 18
And since equation of ellipse with center in `(p,q)` is `((x-p)/a)^2+((y-q)/b)^2=1` we can see that this is ellipse with center in `(-1,1)`.
We will use the following formula to calculate distance from center to focus:
So in our case `f=sqrt(9-6)=sqrt3`.
For directrix we use the following formula:
`x=p pm a^2/(sqrt(a^2-b^2))` sign `pm` is here because ellipse has 2 directrices.
So in our case: `x=-1pm 9/sqrt3=-1pm3sqrt3`
For more on conic sections see link below.
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