Conic Sections - Parabola

Parabola is locus of all points which are equally spaced from a fixed line and a fixed point.

  • The fixed point is called, focus of the parabola.
  • Fixed line is called, directrix of the parabola.Conic Sections - Parabola

In the above figure, F is the focus and line with points A, B and C is the directrix.

According to definition of parabola,

AM = MF
BN = NF
CO = OF

Axis of parabola is a line which is perpendicular to the directrix and passes through the focus of parabola. Vertex of a parabola is the point of intersection of axis and the parabola. Point O is the vertex of the parabola, as shown in the figure below.

Conic Sections - Parabola

Standard Equation

Conic Sections - Parabola

Consider parabola in the given figure,

The simplest form of the equation of a parabola is found when the vertex is at the origin in the coordinate plane.

Let point F(a,0) be focus and O(0,0) be the vertex of the parabola. A and B are two points on the directrix and point P(x,y) is any point on the parabola.

By definition of parabola, O is the midpoint of AF and O is a vertex of the parabola.

That is, AO = OF and coordinates of A will be (-a,0).

Therefore, the equation of the directrix is x+a=0.

Also, FP = AB, (by the definition of parabola)

Since,PB is perpendicular to directrix, coordinates of B is (-a,y)

Using distance formula we get,

\( \sqrt{(x- a)^2 ~+ ~y^2} ~=~ \sqrt{(x~ + ~a)^2} \)

\( (x~-~a)^2~ +~ y^2 ~=~ (x~+~a)^2 \)

\( x^2~ -~2ax ~+~ a^2~ +~ y^2~ =~x^2~+~2ax~+~a^2\)

\(y^2~=~4ax\) —(1)

This is equation for a parabola whose focus is at (a,0)where a > 0.

Equation of Parabola

Equation of the parabola for the figure given below is,

\(y^2\)=-4ax —(2)

Conic Sections - ParabolaEquation of the parabola for the figure given below is,

\(x^2\)=4ay —(3)

Conic Sections - ParabolaEquation of the parabola for figure given below is,

\(x^2\)=-4ay —(4)

Conic Sections - Parabola

The equations (1), (2), (3) and (4) are known as standard equations of a parabola.

Length of Latus rectum

Latus rectum of a parabola is the line segment perpendicular to the axis through focus and its endpoints lie on the parabola.

Length of the latus rectum of the parabola \(y^2~=~4ax\) is given by,

Conic Sections - Parabola

AB is the latus rectum of the above parabola with focus F(a,0).

ABis perpendicular to the X- axis.

Perpendicular distance between the directrix and the focus is,

NF = 2a =AM

By definition of parabola,

AM = AF = 2a

Similarly,

FB = 2a

Therefore,

length of latus rectum = 2a + 2a = 4a

Parabola Examples

Example:

Find equation of directrix, coordinates of focus and length of latus rectum of parabola \(y^2 = 16x\)

Solution:

Comparing given equation \(y^2 = 16x\) with \(y^2 = 4ax\) gives,

4a = 16,a = 4

Co-ordinate of Focus of the parabola is (4,0)

We know, the Equation of directrix is x + a = 0

\(\Rightarrow\) x = – a

Thus x = -4

Length of latus rectum is,

4a = 4 × 4 = 16 units

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