How do you identity if the equation #2x^2+12x+18-y^2=3(2-y^2)+4y# is a parabola, circle, ellipse, or hyperbola and how do you graph it?
Rewrite the equation in General Cartesian Form and then use the conditions of the discriminant to make the determination.
From the reference Conic Sections - General Cartesian Form , the form is:
Rewrite the given equation to match this form:
The discriminant is:
The conditions say that, when the discriminant is less than zero, then the equation represents and ellipse but, when
where x and y correspond to any point
Expand the squares of equation :
To make equation  look more like equation  we divide both sides by 2 and move the constant term to the right side:
We can find the value of h is we set the second term in equation  equal to the second term in equation  equal:
We can do the same for k with the fifth terms in equations  and :
Now that we know the values of h and k, we can substitute the squares into equation  and compute the values of
Write the right side as a square:
To graph equation , set your compass to a radius of 2, make the center point
Here is a graph: