Precalculus / Unit 9: Conic Sections and Parametric Equations
9.5 Identifying Conic Sections
Notes
For $Ax^2 + Cy^2 + Dx + Ey + F = 0$, look at $A$ and $C$:
- Only one of $x^2$, $y^2$ appears (the other coefficient is 0): parabola.
- $A = C$ (same sign, equal): circle.
- $A$ and $C$ have the same sign but are different: ellipse.
- $A$ and $C$ have opposite signs: hyperbola.
- Example: $3x^2 + y^2 - 6 = 0$ is an ellipse.
Practice
What type of conic is $x^2 + y^2 - 4x = 0$?
The $x^2$ and $y^2$ coefficients are equal and have the same sign.
What type of conic is $3x^2 + y^2 - 6 = 0$?
The coefficients 3 and 1 have the same sign but are unequal.
What type of conic is $y^2 - 4x + 2y + 1 = 0$?
Only $y$ is squared, so it is a parabola.
Ready for more? Take the Unit 9 test.