math 2: conic sections Flashcards
conic sections are formed by the intersection of a plane and the 2
nappes of a right cone
a parabola is the set of points that are equidistant from a
given point (focus) and a given line (directrix)
(parabola→x-orientation) it opens to the …. or …
left (p 0)
(parabola→x-orientation) the standard equation is
(y - k)² = 4p(x - h)
(parabola→ x-orientation) the focus is …. and the directrix is ….
(h + p, k); x = h - p
(parabola→ x-orientation) the vertex is
(h, k)
(parabola→ x-orientation) the common distance from the parabola to the focus and directrix is
|p|
(parabola→ y-orientation) it opens .. or …
up (p > 0); down (p
(parabola→ y-orientation) the standard equation is
(x - h)²= 4p(y - k)
(parabola→ y-orientation) the focus is… and the directrix is…
(h, k + p); y = k - p
(parabola→ y-orientation) the vertex is
(h, k)
(parabola→ y-orientation) the common distance from the parabola to the focus and directrix is
|p|
an ellipse is the set of points whose distances from 2 given points (…) sum to a …..
foci; constant
the center of an ellipse … is the intersection of its .. and ….
(h, k); major and minor axes
(ellipse→x-orientation) standard equation is
(x - h)²/a² + (y - k)²/ b² = 1, where a² > b²
(ellipse→x-orientation) the major axis is parallel to the
x- axis
(ellipse→x-orientation) the center is
(h, k)
(ellipse→x-orientation) the vertices are … and …, the endpoints of the … axis
(h-a, k); (h + a, k); major
(ellipse→x-orientation) the major axis is
2a units long
(ellipse→x-orientation)the endpoints of the minor axis are … and ….
(h, k - b); (h, k + b)
(ellipse→x-orientation)the minor axis is
2b units
(ellipse→x-orientation) the foci are on the … axis
major
(ellipse→x-orientation)each focus is at a distance of … from the center, so the foci are at … and …
c = √a² - b²
h - c, k); (h + c, k
(ellipse→y-orientation) standard equation is
(x - h)²/b² + (y - k)²/a² = 1, where a² > b²
(ellipse→y-orientation) the major axis is parallel to the
y-axis
(ellipse→y-orientation) the center is
(h, k)
(ellipse→y-orientation) the vertices are .. and …., the endpoints of the .. axis
(h, k - a); (h, k + a); major
(ellipse→y-orientation) the endpoints of the minor ais are …. and …
(h - b, a); (h + b, a)
(ellipse→y-orientation) the foci are on the … axis. Each focus is at a distance of …. from the center, so the foci are at …., and ….
major; c = √a² - b²; (h, k - c); (h, k + c)
if the larger denominator in an equation of an ellipse is under the x-term, the ellipse has a(n)
x- orientation
if the larger denominator in an equation of an ellipse is under the y-term, the ellipse has a(n)
y-orientation
a hyperbola is the set of points whose distances from 2 fixed points (…) differ by a ….
foci; constant
a hyyperbola has 2 halves, each of which has a … and sides that are … to a pair of intersecting lines
vertex; asymptotic
center of a hyperbola is …, the intersection of its … and … axes
(h, k); transverse; conjugate
(hyperbola→x-orientation) the hyperbola opens to the
sides
(hyperbola→x-orientation) the standard equation is
(x - h)²/a² - (y - k)²/ b² = 1
(hyperbola→x-orientation) the center is
(h, k)
(hyperbola→x-orientation) the vertices are … and ….
(h - a, k); (h + a, k)
(hyperbola→x-orientation) the segment joining the 2 vertices is called the ….. this axis is … and has length…
transverse axis; horizontal; 2a
(hyperbola→x-orientation) the foci are on the…. axis. the distance between the center and each focus is …
transverse; c= √a² + b²
(hyperbola→x-orientation) the foci are .. and …
(h - c, k); (h + c, k)
(hyperbola→x-orientation) the vertical segment through the center with endpoints .. and …, which has length …, is called the … axis
(h, k - b); (h, k + b); 2b; conjugate
(hyperbola→x-orientation) the endpoints of the conjugate axis are not on the
hyperbola
(hyperbola→x-orientation) the equations of the asymptotes are
y - k = ± b/a (x-h)
(hyperbola→y-orientation) the hyperbola opens
up and down
(hyperbola→y-orientation) the standard equation is
(y - k)²/a² - (x - h)²/ b² = 1
(hyperbola→y-orientation) the center is
(h, k)
(hyperbola→y-orientation) the vertices are … and …
(h, k - a); (h, k + a)
(hyperbola→y-orientation) the transverse axis is … and has length …
vertical; 2a
(hyperbola→y-orientation) the foci are … and …
(h, k - c); (h, k + c)
(hyperbola→y-orientation) the conjugate axis is …, has endpoints … and …, and has length ..
horizontal; (h - b, k); (h + b, k); 2b
(hyperbola→y-orientation) the equations of the asymptotes are
y - k = ± a/b (x-h)
if the x-term of the equation is positive, the hyperbola has a(n)
x-orientation
if the y term is positive, the hyperbola has a(n)
y-orientation
eccentricity: measure of an ellipse/ hyperbola’s degree of ….
elongation
the ecentricity of an ellipse is …, which is less than … since …
c/a; 1; c
the eccentricity of a hyperbola is also …, which is greater than … since … in a hyperbola
c/a; 1; c > a
