A fascinating fact about the motions of all heavenly bodies (planets, comets, etc.) is
that they all move in conic sections. Conic sections are the ellipse, parabola, hyperbola,
and the circle . In fact all objects under the influence of gravity moves in conic sections.
Conic sections can be written in mathematics and they are as follows :
I. Circles.
A circle is the set of points in a plane whose
distance from a given fixed point in the plane is constant. The fixed point is the
center of the circle ; the constant distant is the radius .
The equation of a circle of radius centered at the point (h,k) is :
(x-h)2 + (y-k)2 = a2
(a is the radius of the circle.)
II. Parabolas.
A set that consists of all the points in a plane equidistant from a given fixed point
and a given fixed line in the plane is a parabola . The fixed point is the
focus of the parabola. The fixed line is the directrix .
It's standard form is :
y = x2 / 4p
(p is the parabola's focal length.)
III. Ellipses.
An ellipse is the set of points in a plane whose distances from two fixed points
in a plane have a constant sum. The two fixed points are the foci of the
ellipse.
It's standard form is :
x2 / a2 + y2 / b2 = 1
(a>0 , b>0)
(a is the semimajor axis and b is the simiminor axis of the ellipse.)
IV. Hyperbolas.
A hyperbola is the set of points in a plane whose distances from two fixed
points in the plane have a constant difference. The two fixed points are the foci of
the hyperbola.
It's standard form is :
x2 / a2 - y2 / b2 = 1
(a>0 , b>0 , k2 = a2 + b2)