A conic section is the intersection of a plane with a cone. can be defined: as the locus of all points P, such that the ratio of the distance from P to a fixed point (the focus), to the distance from P to a fixed line (the directrix), is constant. The value of this constant is known as the eccentricity, e.
The general form of the equation of a conic section is ax² + 2hxy + by² + 2gx + 2fy + c = 0
If h²<ab, the conic is an ellipse; the plane intersects just one side of the cone at a greater angle than the cone's central angle.
If h²>ab, the conic is a hyperbola; the plane intersects both sides of the cone at a smaller angle than the cone's central angle.
If h²=ab, the conic is a parabola; the plane intersects just one side of the cone at exactly the same angle as the cone's central angle.
Hyperbola
A hyperbola that opens left and right is x²/a² - y²/b² = 1
The transverse axis connects the vertices, and has length 2a.
The conjugate axis is perpendicular to the transverse axis, and has length 2b.
The eccentricity, e, of the hyperbola is defined as b=a sqrt(e²-1), where e>1
Solve for e:
b=a sqrt(e²-1)
b² = a² (e²-1)
(ae)² = a² + b²
ae = sqrt(a² + b²)
e = sqrt(1 + b²/a²)
The vertices are at (-a,0) and (a,0)
The foci are at (-ae,0) and (ae,0)
This can also be expressed (-sqrt(a²+b²),0) and (sqrt(a²+b²),0)
A "rectangular hyperbola" is xy=k
Parabola
The eccentricity, e, of a parabola is defined as 1.
A parabola with a vertical axis of symmetry is y = ax² + bx + c, and a is not equal to zero.
The axis of symmetry of the parabola is x = -b/(2a). The vertex is
(x, ax² + bx + c) =
(-b/(2a), a(-b/(2a))²+b(-b/(2a))+c) =
(-b/(2a), (b²/(4a))-(b²/(2a))+c) =
(-b/(2a), -(b²/(4a))+c) =
(-b/(2a), -(b²-4ac)/(4a))
If the equation of the parabola is rewritten (y-k) = a(x-h)² then the
vertex is (h,k), the focus is (h,k+1/(4a)) and the directrix is y=k-1/(4a).
The latus rectum is the chord of the parabola parallel to the
directrix that passes through its focus, y=k+1/(4a). Since a
parabola is the set of points equidistant from the directrix and focus, then the
point of intersection of the latus rectum and the parabola has distance 1/(2a)
from the directrix and 1/(2a) from the focus. Thus the length of the latus
rectum is 2/(2a)=1/a.
A parabola with its axis parallel to the x axis is x = ay² + by + c, and a is not equal to zero.
The axis of symmetry of the parabola is y = -b/(2a). The vertex is
(ay² +
by + c, y) =
( a(-b/(2a))²+b(-b/(2a))+c, -b/(2a)) =
( (b²/(4a))-(b²/(2a))+c, -b/(2a)) =
( -(b²/(4a))+c, -b/(2a)) =
( -(b²-4ac)/(4a), -b/(2a))
If the equation of the parabola is rewritten (x-h) = a(y-k)² then the
vertex is (h,k), the focus is (h+1/(4a),k) and the directrix is x=h-1/(4a).
The latus rectum is the chord of the parabola parallel to the
directrix that passes through its focus, x=h+1/(4a).
Ellipse
An ellipse with a vertical major axis having its center at (h,k) is
(x-h)²/b² + (y-k)²/a² = 1, where a>b
The eccentricity, e, of the ellipse is defined as b=a sqrt(1-e²), where 0 <= e < 1
Solve for e:
b=a sqrt(1-e²)
b² = a² (1-e²)
(ae)² = a² - b²
ae = sqrt(a² - b²)
e = sqrt(1 - b²/a²)
The length of the major axis is 2a, and the length of the minor axis is 2b.
The foci are at (h,k-ae) and (h,k+ae).
This can also be expressed (h,k-sqrt(a² - b²)) and (h,k+sqrt(a² - b²))
Internet references
Mathwords: foci of an
ellipse, focus of a
parabola, foci of a
hyperbola
Planet Math: conic
section, and the Dandelin sphere
Mudd Math Fun Facts: eccentricity
of conic sections
Mathworld: eccentricity
-- can be interpreted as the fraction of the distance along the semimajor
axis at which the focus
lies.
Related
pages in this website
Cyclic quadrilaterals -- bunch of theorems related to inscribing polygons
in conic sections
