Parabola

What is parabola ??

The parabola is the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix).

Note:The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid.

Some real life examples of parabola.

Path of a object in air

Suspension bridge

Headlight

Satellite dish

Fountain

Video of parabola for everyday life .

More Information about Parabola

Graphical representation of parabola by equation Y=1/2(x-6)(x+2).

A parabola is a curve where any point is at an equal distance from: a fixed point (the focus ), and. a fixed straight line (the directrix ) .

Standard Equation of the Parabola

When the vertex of a parabola is at the ‘origin’ and the axis of symmetry is along the x or y-axis, then the equation of the parabola is the simplest. Here is a quick look at four such possible orientations:

PF = √ {(x – a)2 + y2}

Also, PB = √ (x + a)2

Since, PF = PB [from eq. (1)], we get

√ {(x – a)2 + y2} = √ (x + a)2

So, by squaring both sides, we have (x – a)2 + y2 = (x + a)2 or, x2 – 2ax + a2 + y2 = x2 + 2ax + a2 . Further, by the solving the equation, we have y2 = 4ax … where a > 0. Therefore, we can say that any point on the parabola satisfies the equation:

y2 = 4ax

let’s derive the equation for the parabola shown in Fig.2 (a). As can be seen in the diagram, the parabola has focus at (a, 0) with a > 0. Also, the directrix x = – a. Let F be the focus and l, the directrix. Also, let FM be perpendicular to the directrix and bisect FM at point O. Produce MO to X.

The equations of parabolas in different orientations are as follows:

y2 = – 4ax

y2 = 4ax

x2 = 4ay

x2 = – 4ay.

Latus Rectum

Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose endpoints lie on the parabola as shown below.

Proving length of latus rectum = 4a

Some example of parabola of solving equation

Example: Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola y2 = 8x.

Solution: the parabola equation involves y2, so the axis of symmetry is along the x-axis.

The coefficient of x is positive so the parabola opens to the right. Comparing with the given equation

y2 = 4ax, we find that a = 2.

Thus, the focus of the parabola is (2, 0) and the equation of the directrix of the parabola is x = – 2 . Length of the latus rectum is 4a = 4 × 2 = 8.

Example: Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2).

Solution: Since the vertex is at (0,0) and the focus is at (0,2) which lies on y-axis, the y-axis is the axis of the parabola. Therefore, equation of the parabola is of the form x2 = 4ay.

Thus, we have x2 = 4(2)y, i.e., x2 = 8y

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