Math CentralQuandaries & Queries


Do you have any tips how to identify a conic from its equation?

2x^2 +y^2-12x-4y+22 -y^2+x+4y=1

Best regards

We have two responses for you

Hi Robin.

There are four conics: the circle, ellipse, hyperbola and parabola.

To identify which is which from the equation, first move everything to one side (with a zero on the other side) and simplify as much as possible (this is called "General Form").

Then, examine these things in order:

  1. If only one variable appears squared, then you have a parabola.

  2. If the squared x term and the squared y term are opposite signs (one is positive and one is negative), then you have a hyperbola.

  3. If the squared x term and the squared y term have the same constant multiplier (for example, 3x2 + 3y2), then you have a circle.

  4. The only other choice is an ellipse.

So it comes down to looking at the number and the sign in front of the squared terms.

Hope this helps,
Stephen La Rocque.



When a conic is written in the form Ax2 + By2 + Cx + Dy + E = 0, then the following rules can be used to determine what type of relation it is:

If A = B (not equal to 0), then the conic is a CIRCLE
If A or B is 0 (but not both) then the conic is a PARABOLA
If A and B are both non-zero and have the same sign (+ or - ), then the conic is an ELLIPSE
If A and B are both non-zero and have opposite signs, then the conic is a HYPERBOLA

Hope this helps,

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