A Glimpse At The Conics And The Parabola

To discuss about Parabola, we must know something about the conic first.

A conic is the locus of a point that is moving in a plane in such a way that its distance from a fixed point (focus) bears a constant ratio to its corresponding perpendicular distance from a fixed straight line (directrix). This constant ratio is called the eccentricity (e).

If e = 1, the locus is a Parabola
If e > 1, the locus is a Hyperbola
If e < 1, the locus is an Ellipse

 

History

 

The conic section first came into use when, in the 4th century BC Menaechmus discovered a way to solve the problem of doubling the cube which is impossible to solve with only compass and straightedge. Apollonius of Perga wrote Conic Sections in eight volumes. He discovered several properties of the conic section, and has coined the name “parabola”. The focus - directrix property of the parabola was discovered by Pappus along with many other properties of the conic section

The doubling of a cube means to make a new cube out of a cube whose length of the side is s and volume is V. The new cube has to be such that its volume should be 2V

and length of sides should be This problem is impossible to solve with only compass and straightedge because is not a constructible number.

 

Some terms related to Parabola

 

 

Focus

– It is the fixed point with reference to which the parabola is constructed.

 

 

Directrix

– It is a straight line outside the parabola.

 

 

Axis of symmetry

– It is the line which is perpendicular to the Directrix and passes through the focus. It divides the parabola into two equal halves.

 

 

Vertex – It is the point on the axis of symmetry that intersects the parabola when the turn of the parabola is the sharpest. The vertex is halfway between the Directrix and the focus.

 

 

Focal chord

– It is any chord that passes through the focus.

 

 

Latus rectum

– It is that focal chord which is perpendicular to the axis of symmetry. The latus rectum is parallel to the directrix. Half of the latus rectum is called the semi -latus rectum.

 

 

Focal parameter

– The distance from the focus to the directrix is called the focal parameter.

 

 

The standard form and the vertex form of a parabola

 

A Parabola is represented in the standard form as y = ax2 + bx + c where a is a constant and in the vertex form as y = a(x – h)2 + k , where the coordinates of the vertex is (h, k).

The axis of symmetry is given by the formula x =

y = ax2 + bx + c --- if a > 0, the parabola opens upward like a regular “U”

If a < 0, the parabola opens downward.

y = a(x – h)2 + k ----- If a is positive, the parabola opens upward like a regular “U”

If a is negative, the parabola opens downward.

The graph of the parabola widens i.e., it stretches sideways if |a| > 1 and the opposite happens i.e., it narrows if |a| < 1.

Parabolas can open up, down or in any other direction. They can be repositioned and rescaled to fit exactly on another parabola. So we can say that all parabolas are similar.

 

Uses of parabolas

 

Parabolas have many uses. They are used in automobiles, in hitting targets with missiles etc. We can see the vast use of parabolas in physics, and in engineering.