Various types of conics sections

Definition of conic sections: We can define conic section as follows: Consider a double cone. If we have a plane that cuts this double cone, the cross section thus obtained at the intersection of the plane and the cone is called a conic section.

Depending on the angle at which the plane cuts the cone and the position of the plane, we can have mainly 4 different conics: circle, ellipse, parabola and hyperbola.

Conic s have been studied for over 2000 years. Greek mathematician Apollonius studied them intensely. He wrote a book ‘the conic’ that remained a standard work on the topic for eighteen centuries. In the sixteenth century, Galileo declared that the trajectory of a projectile was a parabola. The reflectors in head lights of a car, the speakers in a sound system and the mirrors in a telescope are all in the shape of a parabola.

Parabolic mirrors are used to harness solar energy. In the seventeenth century, Kepler declared that planets revolved around the sun in elliptical orbits. It is because of our knowledge of ellipses that precise predictions of time and place of solar and lunar eclipses is possible. The path of motion of a comet in the solar system is also in the shape of a parabola, ellipse or a hyperbola. Knowledge of conics is extremely useful in such terrestrial sciences as architecture and bridge building.

Thus, the study of conics has been proved very useful in space sciences, mechanics, optics, engineering, architecture and other fields.

How to graph conic sections?
To be able to graph conics, let us look at the following concept. Suppose line l is a fixed vertical line and another line m intersects l in the point V and makes an angle of measure a(0 < a < pi/2) with it. If m is made to rotate around V in such a way that a remains constant, then the surface generated is called a double cone. The point V is the vertex and the line m is a generator of the double cone. The line l is the axis of the double cone.  Then the plane cutting this double cone defines the four conics as follows: (see picture below)

Solving Solid Geometry

In solid geometry we study three dimensional geometry (3-D geometry).
For examples: Cube, cuboid, cylinder, cone, sphere, Pyramids, Prisms etc. Dimensions are the terms as length, width, height, thickness etc. A three dimensional figure must have length, width and height.

Cube : A three dimensional shape having equal length(a), width(a) and height (a).
Cuboid: A three dimensional shape having different length(l), width(w) and height(h).
Cylinder: A three dimensional shape having two circular faces of radius(r) at two ends and a curved surface of height(h).
Cone: A three dimensional shape having a circular face at one end and a curved surface of height(h).
Sphere: A three dimensional shape of radius r. For example: a ball.

Formulas for Solving Solid Geometry
(1) Cube :           Lateral Surface Area ( Area of four sides i.e. front, back, left, right ) = 4a2
                           Total Surface Area( Area of all six faces) = 6a2
                            Volume = a x a x a = a3
(2) Cuboid:        Lateral Surface Area ( Area of four sides i.e. front, back, left, right ) = 2h(l+w)
                            Total Surface Area( Area of all six faces) = 2( lw + wh + hl )
                             Volume = lwh
(3) Cylinder:      Curved surface area = 2Ï€rh
                            Total surface area (including two circles on both ends) = 2Ï€r(r+h)
                            Volume = 2Ï€r2h
(4) Cone :          Curved surface area = Ï€rl where l is the slant height of the cone l = `sqrt(h^2 + r^2)`
                            Total surface area (including a circles on the base) = Ï€r(r+l)
                            Volume = 1/3 Ï€r2h
(5) Sphere :      Surface area = 4Ï€r2
                            Volume = 4/3 Ï€r3

How to Solve Problems for Solid Geometry
Step 1) Make a figure of solid given in the problem.
Step 2) Write the dimensions of the solid e.g. length, width, height, radius etc.
Step 3) Apply the formula for particular solid geometry figure.
Step 4) Write the unit of the particular physical quantity e.g. square meters, cubic centimeters etc.

Circles and semi circles

Important definitions related to circles:

1. Circle: A circle is a simple closed curve all of whose points are at a constant distance from a fixed point in the same plane. The fixed point is called the centre of the circle.

2. Circumference of a circle: The distance right around the circle is called its circumference. It is the perimeter of the circle. The traditional method to measure this perimeter of a circle was using a thread or a rope long the circumference. However this method is not too practical for very large circular fields or pieces of land. Therefore for all practical purposes, the following formula was derived by mathematicians for circumference of a circle.
 C = pi*d. Where, C = circumference of the circle, d = diameter of the circle and pi = ratio of the circumference of a circle to its diameter. The value of the Greek letter pi (read as pi) was experimentally calculated by mathematicians. It is an irrational number. A decimal of nonrecurring type. It is a constant. Indian mathematician Ramanujan gave two approximations for the value of pi in the year 1914. Generally all mathematicians have accepted the value of this constant as 3.141 592 653 589 793....

3. Semi circle: A diameter divides a circle into two equal parts which are called semi circles. The length of the curved portion of a semi circle is equal to half the circumference of the circle. The total perimeter of a semi circle is equal to sum of half the circumference and diameter. So putting that mathematically, perimeter of a semi circle = P,
P = C/2 + d, where C = circumference of a circle with diameter d.
From any point on the semi circle if we draw two lines that meet both the ends of the diameter, the angle so formed is called angle in a semi circle. This angle in a semi circle is always a right angle.

4. Unit of a circle (or unit circle): A circle which has a unit radius is called a unit circle. In other words a circle with radius = 1 and diameter = 2 is called a unit circle.

5. Intersecting circles: If there are two circles in a plane then any of the following three possibilities are there:
(a) The circles do not touch or intersect each other at all.
(b) The circles touch each other in exactly one point.
(c)  The circles intersect each other in exactly two points.