Conic Sections Parabola y d1 = d2 Parabola: The set of all points in a plane equidistant (x,y) from a fixed point (focus) and a line (directrix) d1 focus You can identify a parabola from the general form of a p d2 conic section because only one variable will be squared and the other variable will be linear. vertex x y=-p directrix x 2 4 py Ellipse y d1 + d2 = 2a Ellipse: The set of all points in a plane such that the b sum of the distances from two fixed points (foci) is constant. x e -a -c c t a r That constant is the length of the major axis. focus e x d1 v d2 In the general form of a conic section, the squared terms -b will be the same sign. A circle is a special case of the (x,y) ellipse that occurs when the coefficients on the squared terms are the same. x2 y 2 1 Hyperbola x2 y 2 y |d1 - d2| = 2a Hyperbola: The set of all points in a plane such that the difference of the distances from two fixed b points (foci) is constant. -c -a vertex c focus a x d2 d1 (x,y) That constant is the length of the transverse axis. -b In the general form of a conic section, the squared terms will have different signs. x2 y 2 1 x2 y 2 Conic Sections Parabola y y The starting point is the vertex. y r t e f There is an axis (of symmetry) x o m e s which contain the focus and t s i m r u x y e c axis of a s the vertex and is v (h,k) o f symmetry perpendicular to the directrix. -p p Move p units along the axis of focus p (h,k) symmetry from the vertex to x vertex i r the focus. t x c x e Move -p units along the axis of r -p directrix i d symmetry from the vertex to the directrix (which is a line). (x h)2 4p(y k) (y k)2 4p(x h) The focus is within the curve. Ellipse y endpoint of y The center is the starting point. minor axis The major axis contains the foci b focus c and vertices. focus a Major axis length = 2a c (h,k) vertex endpoint of (h,k) Minor axis length = 2b minor axis b Distance between foci = 2c The foci are within the curve. Since the vertices are the furthest x a x away from the center, the vertex pythagorean relationship is ... (x h)2 (y k)2 (y k)2 (x h)2 a2 b2 c2 1 1 a2 b2 a2 b2 Hyperbola y y focus The center is the starting point. endpoint of c The transverse axis contains the s conjugate axis i x foci and vertices. f a b o Transverse axis length = 2a e t focus vertex t n a i Conjugate axis length = 2b g c a (h,k) o (h,k) u p b j Distance between foci = 2c d n n o The foci are within the curve. e a c Since the foci are the furthest x vertex x away from the center, the pythagorean relationship is ... (x h)2 (y k)2 (y k)2 (x h)2 c2 a2 b2 1 1 a2 b2 a2 b2 y The equations of the asymptotes are y k x h x.