Chapter 9 Conic Sections

Conic Sections (1) Circle A circle is formed when a plane is perpendicular to the axis of the cones.

Conic Sections (2) Ellipse An ellipse is formed when a plane cuts only one of the cones, but is neither perpendicular to the axis nor parallel to the base.

Conic Sections (3) Parabola A parabola is formed when a plane is parallel to a slant height of the cone.

Conic Sections (4) Hyperbola A hyperbola is formed when a plane intersects both cones, but does not pass through the common vertex.

9.1 The Ellipse An ellipse is the set of all points, P, in a plane the sum of the whose distances from two fixed points, F1 and F2, is constant. These two fixed points are called the foci. The midpoint of the segment connecting the foci is the center of the ellipse. P’(x,y) P’’(x,y)

Let PF1+PF2 = 2a where a > 0

standard equation of an ellipse

major axis = 2a vertex minor axis = 2b length of semi-major axis = a length of the semi-minor axis = b

AB major axis CD minor axis A and B vertices O center F focus

Other forms of the equation of an Ellipse Vertical Ellipse where a2 – b2 = c2 and a > b > 0

y (h, k) x O

y (h, k) x O