Math CentralQuandaries & Queries


Question from Donna, a parent:

What type of conic section is 3x² + 3y² - 4y - 8 = 0


There are both x2 and y2 terms so it's not a parabola. The fact that the coefficients of the x2 and y2 terms are both positive means it's not a hyperbola. Since the coefficients of both the x2 and y2 terms are the same (they are both 3) it's probably a circle. All that remains is to put it in standard form

(x - h)2 + (y - h)2 = r2

to check that the right side is positive. If the right side is zero then the radius of the circle is zero and the conic is just a point. If the right side is negative then there are no points on the graph of 3x² + 3y² - 4y - 8 = 0.

To put the equation in standard form first divide through by 3 to get

x2 + y2 - 4/3 y - 8/3 = 0

Now complete the square of the expression y² - 4/3 y - 8/3 = 0. This will allow you to write

x2 + y2 - 4/3 y - 8/3 = 0

in the form

(x - 0)2 + (y - k)2 = ?

If the right side is positive then you have a circle.

I hope this helps,

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