Conic Sections - Parabola

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 of a parabola

Conic Sections - Parabola

Consider parabola in the given figure,

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

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

By definition of parabola, O is the mid point of AF and O is vertex of the parabola.

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

Therefore, equation of the directrixis 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.

Important points to note:,

  • Equation of the parabola for figure given below is,

\(y^2\)=-4ax —(2)

Conic Sections - Parabola

  • Equation of the parabola for figure given below is,

\(x^2\)=4ay —(3)

Conic Sections - Parabola

  • Equation 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 parabola.

Length of Latus rectum

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

Length of 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

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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Practise This Question

Kushagra is drawing a railway track on paper as a part of his project. He asked Sarosh to draw a line parallel to the given line. Sarosh said that we can only construct a line parallel to the given line using alternate angles concept. Is this true?