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Parabola

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When a solid is cut by a plane, the curve common to the solid and the plane i.e., the curve which lies on the surface of the solid and the plane is called "section of the solid by a plane." Similarly, when a right circular cone is cut by planes in different positions, the sections obtained are the curves-circle, parabola, ellipse and hyperbola.


Conic Section
In the figure, $O$ is the vertex of the solid cone and $AOA$' is its axis. a is the semi vertical angle. The section of the cone by a plane perpendicular to its axis is the circle. When a conic is cut by a plane which makes an angle ß = a with the axis then the section formed is called as Parabola.

What is Parabola?

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Parabola is the locus of a point which moves such that its distance from the focus is equal to its distance from the directrix.
It appears to be in the form of arch, when inverted into an arch structure, it results in a form, which allows equal vertical loading along its length. A parabola is a graph of a quadratic equation.

Properties of Parabola

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Mainly the description of a parabola involves a point that is called as focus and a line called directrix. In simple word, directrix is a path followed by a point or line when moving. The line perpendicular to the directrix and which passes through the focus is called as axis of symmetry. The intersection point of parabola and axis of symmetry is called as vertex. The cord of parabola which is parallel to the directrix and passes through the focus is called as latus rectum.


Properties of Parabola



Equation of a Parabola

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A parabola is the set of all points $(x, y)$ in a plane that are equidistant from a fixed line and a fixed point not on the directrix.

In the given figure below, $S$ is the focus and $l$ is the directrix of the parabola. Draw $SZ$ perpendicular to the line l. Let ''$O$'' be the middle point of $SZ$. Take $O$ as the origin, $OS$ produced as the x-axis and $OY$ perpendicular to $OS$. Let $P(x, y)$ be any point on the curve. Join $PS$ and draw $PM$ perpendicular to directrix $l$ and $PN$ perpendicular to x-axis. Take $OS$ = $OZ$ = $a$. Then the co-ordinates of $S$ are $(a, 0)$ and that of $Z$ = $(- a, 0)$

Equation of Parabola


Here $S$ = $(a, 0)$ and $P$ = $(x, y)$. So by distance formula

$PS$ = $\sqrt{(x - a)^2 + (y - 0)^2}$

$PM$ = $NZ$ = $NO$ + $OZ$ = $x + a$.

[ Because NO is the x- coordinate of the point $P(x, y)$].

By definition, $\frac{PS}{PM}$ = 1 or $PS$ = $PM$

$\therefore$ $\sqrt{(x-a)^2+y^2}$ = $(x + a)$.

Square both sides,

$(x - a)$2 + $y$2 = $(x + a)$2

i.e., $x$2 + $a$2 - 2$ax$ + $y$2 = $x$2 + $a$2 + 2$ax$

i.e., $y$2 = 4$ax$

This is the equation of the Parabola with vertex as origin and the x-axis as its axis of symmetry.

The standard form of the equation of parabola with vertex at the origin and the y-axis as its axis of symmetry is derived in a same manner.

Latus Rectum of Parabola

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Through the focus $S$ as given in the figure below, draw a line perpendicular to the axis of the Parabola. Let the line cut Parabola at $L$ and $L'$, then $LSL$' is called the Latus rectum of the parabola. It can be regarded as the double ordinate passing through the focus. Draw $LM$ perpendicular to the directrix.

Parabola Latus Rectum

Then by the difinition $LS$ = $LM$ = $SZ$ = 2$a$.

$\therefore$ the lenght of semi latus rectun is $LS$ = 2$a$.

Length of latus rectum is $LSL$' = 2$LS$ = 2(2$a$) = 4$a$.

Vertex of a Parabola

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The vertex of the parabola is the point where the parabola crosses its axis. It is the highest or lowest point or also known as the maximum and minimum of vertex. When the co-efficient of the x2 term is positive, then the vertex will be the lowest point on the graph, i.e. the point will be lying at the bottom of the 'U' shape and when the co-efficient of the x2 term is negative, the vertex will be in the highest at the top of 'U' shape.

Vertex of Parabola

Focus and Directrix of Parabola

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A parabola is the locus of a point which moves in a plane so that its distance from a fixed point in a plane is equal to its distance from a fixed straight line in that plane.

Focus and Directrix of Parabola

The fixed point is called as the focus and the fixed straight line is called the directrix of the parabola. The distance of any point on the parabola from the focus is called the focal distance of the point.

Example of Parabola

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Below are the few examples based on Parabola

Question:

Show that $(y - 3)$2 = 12$(x - 1)$ is a parabola. Find the equation of its axis and co-ordinates of the vertex and focus.

Solution:

Given: $(y - 3)$2 = 12$(x - 1)$

Change the origin to the point (1, 3) that is [$h$ = 1, $k$ = 3]

then $x$ = $X$ + 1, $y$ = $Y$ + 3

($Y$ + 3 - 3)2 = 12( $X$ + 1 - 1)

i.e., $Y$2 = 12 $X$ which is the equation of the parabola.

The Axis of the Parabola is $y$ = $k$

i.e., $y$ = 3 is a line parallel to x- axis.

$\therefore$ Vertex = (1, 3)