`color{blue}(★ ul"Parabola")`

`\color{green} ✍️` A `color{blue}(ul"parabola")` is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane.

`\color{green} ✍️` The fixed line is called the `color{blue}(ul"directrix")` of the parabola and the fixed point `F` is called the `color{blue}("focus")` (`Fig \ \11.13`).

`\color{green} ✍️` `color{blue}(ul"Para")` means `"for"` and `color{blue}(ul"bola")` means `"throwing"`, i.e., the shape described when you throw a ball in the air .

`\color{green} ✍️` A line through the focus and perpendicular to the directrix is called the `color{blue}(ul"axis of the parabola")`.

`\color{green} ✍️` The point of intersection of parabola with the axis is called the `color{blue}(ul"vertex of the parabola.")` ( `Fig \ \11.14`).

`\color{green} ✍️` The equation of a parabola is simplest if the vertex is at the origin and the axis of symmetry is along the `x-`axis or `y-`axis. The four possible such orientations of parabola are shown below in`Fig \ \ 11.15` `(a)` to `(d).`

`\color{green} "We will derive the equation for the parabola shown above in Fig 11.15 (a)"` with focus at `(a, 0) a > 0;` and directricx `x = – a` as below:


Let `F` be the focus and `l` the directrix. Let FM be perpendicular to the directrix and bisect FM at the point `O`. Produce `MO` to `X`.

By the definition of parabola, the mid-point `O` is on the parabola and is called the vertex of the parabola.

Take `O` as origin, OX the x-axis and `OY` perpendicular to it as the `y-`axis.

Let the distance from the directrix to the focus be `2a.` Then, the coordinates of the focus are `(a, 0)`, and the equation of the directrix is `x + a = 0` as in Fig11.16. Let `P(x, y)` be any point on the parabola such that `PF = PB,`



where `PB` is perpendicular to `l`. The coordinates of `B` are `(– a, y).` By the distance formula, we have

`PF = sqrt((x-a)^2 +y^2)` and `PB = sqrt((x+a)^2`

Since `PF = PB,` we have

`sqrt((x-a)^2 +y^2) = sqrt((x+a)^2`

i.e. `(x – a)^2 + y^2 = (x + a)^2`

or `x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2`

or `y^2 = 4ax ( a > 0).`

Hence, any point on the parabola satisfies

`y^2 = 4ax.`

Conversely, let `P(x, y)` satisfy the equation (2)

`PF = sqrt((x-a)^2 +y^2) = sqrt((x-a)^2 + 4ax)`

`= sqrt[(x+a)^2] = PB`

and so `P(x,y)` lies on the parabola.

Thus, from (2) and (3) we have proved that the equation to the parabola with vertex at the origin, focus at (a,0) and directrix `x = – a` is `y^2 = 4ax.`

Discussion In equation `(2),` since `a > 0, x` can assume any positive value or zero but no negative value and the curve extends indefinitely far into the first and the fourth quadrants. The axis of the parabola is the positive x-axis. Similarly, we can derive the equations of the parabolas in:


Fig 11.15 (b) as `y^2 = – 4ax,`
Fig 11.15 (c) as `x^2 = 4ay,`
Fig 11.15 (d) as `x^2 = – 4ay,`

These four equations are known as `color{green}"standard equations of parabolas."`

`color{blue} (★ ul"Latus rectum")`

Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola (Fig11.17).

`color{blue} (★ ul"To find the Length of the latus rectum of the parabola" \ \ y^2 = 4ax)` (Fig 11.18).

By the definition of the parabola, `AF = AC.`

But `AC = FM = 2a`

Hence `AF = 2a.`

And since the parabola is symmetric with respect to `x`-axis `AF = FB` and so

`AB =` `color{blue} (ul"Length of the latus rectum" = 4a.)`

Various terms related to parabola are explained in the image opposite.

`color{blue}{ulmathtt"From the standard equations of the parabolas, we have the following observations:"}`

`color{blue}(☛)` Parabola is symmetric with respect to the axis of the parabola.

If the equation has a `y^2` term, then the axis of symmetry is along the `x`-axis and if the equation has an `x^2` term, then the axis of symmetry is along the y-axis.

`color{blue}(☛)` When the axis of symmetry is along the `x`-axis the parabola opens to the
`\ \ \ \ \ \ \ \ \ \(a)` right if the coefficient of `x` is positive,
`\ \ \ \ \ \ \ \ \ \(b)` left if the coefficient of `x` is negative.

`color{blue}(☛)` When the axis of symmetry is along the y-axis the parabola opens
`\ \ \ \ \ \ \ \ \ \(c)` upwards if the coefficient of `y` is positive.
`\ \ \ \ \ \ \ \ \ \(d)` downwards if the coefficient of `y` is negative.