# 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?

##### 1 Answer

Rewrite the equation in General Cartesian Form and then use the conditions of the discriminant to make the determination.

#### Explanation:

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 [3]:

To make equation [2] look more like equation [4] we divide both sides by 2 and move the constant term to the right side:

Add

We can find the value of h is we set the second term in equation [4] equal to the second term in equation [6] equal:

We can do the same for k with the fifth terms in equations [4] and [6]:

Now that we know the values of h and k, we can substitute the squares into equation [6] and compute the values of

Write the right side as a square:

To graph equation [8], set your compass to a radius of 2, make the center point

Here is a graph: