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**Definition:**

A parabola is the locus of a point, whose distance from a fixed point (focus) is equal to perpendicular distance from a fixed straight line (called directrix).

**Some Standard Forms of Parabola**

For

(a) Vertex is

(b) Focus is

(c) Axis is

(d) Directrix is

**Focal Distance:**

The distance of a point on the parabola from the focus.

**Focal Chord:**

A chord of the parabola, which passes through the focus.

**Double Ordinate:**

A chord of the parabola, which is perpendicular to the axis of the symmetry.

**Latus Rectum:**

A double ordinate passing through the focus or a focal chord perpendicular to the axis of the parabola is called the latus rectum. (L.R)

For

length of L.R

Ends of L.R are and

**Note:**

(i) Perpendicular distance from the focus on the directrix = half the latus rectum.

(ii) Vertex is the middle point of the focus and the point of intersection of directrix and axis.

(iii) Two parabolas are said to be equal if they have the same Latus Rectum.

**Key Point:**

The equation of a parabola with directrix and focus is .

The parametric equation of a parabola with directrix and focus is .

Example:

Find the equation of the parabola whose focus is at and the directrix is .

Answer:

Let be any point on the parabola whose focus is and the directrix

This is the equation of parabola.

**Parametric form of Parabola:**

(i) For

(ii) For

(iii) For

(iv) For

Example:

Find the parametric equation of the parabola

Solution:

Given

(i)

(ii)

From (i)

From (ii)

**Position of a Point Relative to a Parabola:**

The position of a point defined by the point lies outside, inside or on the parabola according as the expression is positive, negative or zero.

Example:

Check the point situated to the parabola

Solution:

lies outside the parabola

**Line and a Parabola:**

The line meets the parabola in two points real, coincident or imaginary according as a respectively.

Length of the chord intercepted by the parabola on the line is :

**Note:**

The equation of a chord joining and is

If and are the ends of a focal chord of the parabola then . Hence the co-ordinates at the extremities of a focal chord can be taken as and

Length of the focal chord making an angle with the -axis is

**Tangent to the Parabola **

An equation of the tangent at a point parabola can be obtained by the replacement method or derivatives.

In the replacement method, the following changes are

So, the tangents are:

(i) at the point

(ii) at

(iii) at

(iv) Point of intersection of the tangents at the point and is

**Normal to Parabola:**

Normal is obtained using the slope of the tangent.

Slope of tangent at

Slope of normal

(i) at

(ii) at

(iii) at

**Note:**

Point of intersection of normals at and is

If the normals to the parabola at the point , meets the parabola again at the point , then

If the normals to the parabola at the points and intersect again on the parabola at the point , then and the line joining and passes through a fixed point

**Application of Parabola:**

A satellite dish receiver is the shape of a Parabola.

An arched underpass of a road has the shape of a Parabola.

A soup bowl has a cross-section with a parabolic shape.

**Some Important Points:**

(i) The tangent and the normal at a point on the parabola are the bisectors of the angle between the focal radius and the perpendicular from that fixed point on the directrix.

From this, we conclude that all the rays emitting from the focus will remain parallel to the axis of the parabola after reflection.

(ii) The portion of the tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.

(iii) Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.

(iv) Semi Latus Rectum of the parabola , is the harmonic mean between segments of any focal chord of parabola.

(v) The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.

(vi) Length of subnormal is constant for all points on the parabola and is equal to the semi-latus rectum.

**Comparative Study of Four Parabolas:**

Parabola | |||||

1 | Focus | ||||

2 | Directrix | ||||

3 | Axis | -axis | axis | -axis | -axis |

4 | Vertex | Origin | Origin | Origin | Origin |

5 | Latus Rectum | ||||

6 | Ends of Latus Rectum | ||||

7 | Lies in | I and IV quadrants | I and III quadrants | I and II quadrants | III and IV quadrants |

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