<aside> 💡 The ONLY terms to consider are the coefficients of $x^2$ and $y^2$.

</aside>

The following questions must be done in 5 seconds, so it is essential to be quick at identifying the coefficients and signs and what these mean:

  1. A positive and negative coefficient for the $x^2$ and $y^2$ terms = Hyperbola
  2. Same coefficients = Circle
  3. Different and positive coefficients = Ellipse
  4. A missing x^2 or y^2 term (only one of the two) = Parabola

Example 1 (Toss-Up):

$$ \text{Identify the type of conic section represented by the equation} \space x^2-16y^2=64 $$

Since the coefficients A and C are NOT equal, this is not a circle. Since both "A" and "C" are present, it is not a parabola. When you see two different coefficients for $x^2$ and $y^2$, it is immediately either an ellipse or a hyperbola. Since one of the coefficients is (-), it is automatically a hyperbola. If it were $x^2+16y^2=64$, then it would be an ellipse.

Example 2 (Toss-Up):

$$ \text{Identify the type of conic section represented by the equation}\space x^2+y^2+10y+16 $$

Remember that only the $x^2$ and $y^2$ terms matter. Since they have the same coefficient (1), it is automatically a circle.

Practice Problems:

  1. The graph of the following equation is an example of what conic section: $2x^2=2y^2+13$
  1. What type of conic section is represented by the following equation: $0 = x^2 + 2x – y^2 + 6y – 14$