4.5 Conic Sections

Conics from equations

Conic sections (circles, ellipses, parabolas, hyperbolas) come from slicing a cone with a plane, but on the exam you'll mostly identify and graph them from equations. Standard forms center on (h, k): circles use x²+y²=r², ellipses/hyperbolas use x²/a² and y²/b² terms with a plus (ellipse) or minus (hyperbola) sign.

Key features to identify

For ellipses: center, vertices, co-vertices, foci along the major axis. For hyperbolas: center, vertices, foci, and asymptotes (slopes ±b/a or ±a/b depending on orientation). For parabolas: vertex, focus, and directrix, with axis of symmetry determined by which variable is squared.

Parametric and polar connections

Conics can also be represented parametrically (e.g., x = h + a cos t, y = k + b sin t for an ellipse) or as polar equations using r in terms of θ, especially useful for conics with a focus at the origin. Recognizing eccentricity (e) helps classify: e=0 circle, 0<e<1 ellipse, e=1 parabola, e>1 hyperbola.

Key terms

Standard form of an ellipse (center h,k)

(x−h)²/a² + (y−k)²/b² = 1, where a and b are the semi-axis lengths.

Standard form of a hyperbola (center h,k)

(x−h)²/a² − (y−k)²/b² = 1 (or with y first), featuring asymptotes and two branches.

Eccentricity (e)

A value classifying conics: e=0 circle, 0<e<1 ellipse, e=1 parabola, e>1 hyperbola.

Asymptotes of a hyperbola

Lines the hyperbola approaches but never touches; slopes are ±b/a (horizontal transverse axis) or ±a/b (vertical).

Parabola key features

Defined by a vertex, focus, and directrix; every point on the parabola is equidistant from the focus and directrix.

Try an AP-style question

What are the center and radius of (x − 1)² + (y + 2)² = 9?

1. Center (1, −2), radius 9
2. Center (−1, 2), radius 3
3. Center (1, 2), radius 3
4. Center (1, −2), radius 3