1 Reference for Chapter 11: Conic Sections Parabola Ellipse Hyperbola y = 1 4 p x 2 p<0: opens down x 2 a 2 + y 2 b 2 = 1 (a > b > 0) x 2 a 2 − y 2 b 2 = 1 Standard equation (center at origin) x = 1 4 p y 2 p<0: opens left x 2 b 2 + y 2 a 2 = 1 (a > b > 0) y 2 a 2 − x 2 b 2 = 1 |p| or a/b incr. opens wider gets flatter gets flatter Focus / Foci 1 2 2 Vertex / Vertices 1 2 2 Directrix 1 n/a n/a Asymptotes no no 2 Axis / Axes 1 2: major, minor 2: transverse, conjugate Parabola: the set of all points in a plane equidistant from a fixed point F (the focus) and a fixed line l (the directrix) that lie in the plane. (Sec. 11.1) The axis of the parabola is the line through F that is perpendicular to the directrix. The vertex of the parabola is the point V on the axis halfway from F to l. Ellipse: the set of all points in a plane, the sum of whose distances from two fixed points in the plane F and F’ (the foci) is a positive constant. (Sec. 11.2) The midpoint of the segment F’F is the center of the ellipse. The points V and V’ are the vertices of the ellipse, and the line segment V’V is the major axis. The segment M’M is the minor axis. A circle is the set of points in a plane whose distance from a single point in the plane, its center, is a constant; thus, a circle can be considered a special case of an ellipse in which the foci coincide. Hyperbola: the set of all points in a plane, the difference of whose distances from two fixed points in the plane (the foci) is a positive constant. (Sec. 11.3) The midpoint of the segment F’F is the center of the hyperbola. The points V and V’ are the vertices of the hyperbola, and the line segment V’V is the transverse axis. The segment W’W is the conjugate axis. The dotted lines are the hyperbola’s asymptotes.

Reference for Chapter 11: Conic Sectionshomes.soic.indiana.edu/donbyrd/Teach/CAMMHS/Reference_Chap11... · 1 Reference for Chapter 11: Conic Sections Parabola Ellipse Hyperbola €

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Reference for Chapter 11: Conic Sections

Parabola Ellipse Hyperbola

€

y =14 p

p<0: opens down

€

a2+y2

b2= 1

(a > b > 0)

€

a2−y2

b2= 1

Standard equation

(center at origin)

€

x =14 p

p<0: opens left

€

b2+y2

a2= 1

(a > b > 0)

€

a2−x2

b2= 1

|p| or a/b incr. opens wider gets flatter gets flatter

Focus / Foci 1 2 2

Vertex / Vertices 1 2 2

Directrix 1 n/a n/a

Asymptotes no no 2

Axis / Axes 1 2: major, minor

2: transverse, conjugate

Parabola: the set of all points in a plane equidistant from a fixed point F (the focus) and a fixed line l (the directrix) that lie in the plane. (Sec. 11.1)

The axis of the parabola is the line through F that is perpendicular to the directrix. The vertex of the parabola is the point V on the axis halfway from F to l.

Ellipse: the set of all points in a plane, the sum of whose distances from two fixed points in the plane F and F’ (the foci) is a positive constant. (Sec. 11.2)

The midpoint of the segment F’F is the center of the ellipse. The points V and V’ are the vertices of the ellipse, and the line segment V’V is the major axis. The segment M’M is the minor axis.

A circle is the set of points in a plane whose distance from a single point in the plane, its center, is a constant; thus, a circle can be considered a special case of an ellipse in which the foci coincide.

Hyperbola: the set of all points in a plane, the difference of whose distances from two fixed points in the plane (the foci) is a positive constant. (Sec. 11.3)

The midpoint of the segment F’F is the center of the hyperbola. The points V and V’ are the vertices of the hyperbola, and the line segment V’V is the transverse axis. The segment W’W is the conjugate axis. The dotted lines are the hyperbola’s asymptotes.