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 **2***V*

and length of sides should be *T*his 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 = ax****2**** + 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 = ax****2**** + 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.