Conic sections

Telling them apart

Look at the squared terms. Only one squared: parabola. Both squared and added with the same coefficient: circle. Both added with different coefficients: ellipse. Subtracted: hyperbola.

Standard forms

Circle (x−h)² + (y−k)² = r². Ellipse (x−h)²/a² + (y−k)²/b² = 1. Hyperbola is the same with a minus.

Getting to standard form

Complete the square in x and in y separately. That is the whole technique, and it is why completing the square is worth knowing cold.

Which way a hyperbola opens

Whichever variable is positive. x² first means it opens left and right, y² first means up and down.