Parabola

The locus of a point which is equidistant from a fixed point (called the focus) and a fixed line (called the directrix) is called a parabola.

Equation of a Parabola (Standard Form)

Let $S$ be the focus and $ZM$ be the directrix of the parabola. $SZ$ is drawn perpendicular to $ZM$. Let $A$ be the middle point of $SZ$ so that $SA=AZ$. Then, $A$ is the vertex and $ZAS$ is the axis of the parabola.

To determine the equation of a parabola in the standard form, take the vertex $A$ at the origin, the focus $S$ on the x-axis so that the axis of the parabola is the x-axis and the directrix is parallel to y-axis.

Equation of a Parabola (Standard Form)

Let $AS=a$. Then the coordinates of $Z$, $A$ and $S$ are $(-a,0)$, $(0,0)$ and $(a,0)$ respectively. The equation of the directrix is $\:x+a=0$. [Equations of Straight Lines Parallel to the Axes (Some Fundamental Formulae)]

Let $P(x,y)$ be any point on the parabola. Join $PS$ and draw $PM\perp ZM$. Then, from the definition of parabola, \[PS=PM\] \[PS^2=PM^2\] \[(x-a)^2+y^2=(x+a)^2\] \[\therefore y^2=4ax\]

This is the standard form of a parabola.

Special Cases

If $a$ is positive, $y^2=4ax$ is a parabola whose axis is the x-axis and the curve lies to the right side of the y-axis. If $a$ is negative, $y^2=-4ax$ lies to the left side of the y-axis.

Parabola whose axis is the x-axis: y^2=4ax and y^2=-4ax

Similarly, if $a$ is positive, $x^2=4ay$ is a parabola whose axis is the y-axis, focus $(0,a)$ and directrix $\:y+a=0$ and the curve lies above the x-axis. If $a$ is negative, $x^2=-4ay$ lies below the x-axis.

Parabola whose axis is y-axis: x^2=4ay and x^2=-4ay

Parameter

The point $(at^2,2at)$ clearly satisfies the equation of the parabola $y^2=4ax$. Here, $t$ is called the parameter. Hence, by the points $t_1$ and $t_2$ on the parabola, we mean the points $(at_1^2,2at_1)$ and $(at_2^2,2at_2)$.

Focal Distance, Focal Chord and Latus Rectum

Consider a parabola whose vertex is $A$ and focus is $S$. Let $P$ be any point on the parabola.

Focal distance, focal chord and latus rectum of a parabola

Focal Distance

The distance of any point on the parabola from the focus is called the focal distance or focal radius of the point. In the figure, $SP$ is the focal distance and $SP=PM=x+a$.

Focal Chord

Any chord of the parabola passing through the focus is called focal chord. In the figure, $PSP’$ is the focal chord of the parabola.

Latus Rectum

The chord of the parabola passing through the focus and perpendicular to the axis is called latus rectum. In the figure, $LSL’$ is the latus rectum. $L$ and $L’$ are the end points of the latus rectum which are the points of intersection of the parabola $y^2=4ax$ and the line $x=a$. By solving these two equations, we get $x=a$ and $y=\pm 2a$.

Hence, the coordinates of the end points $L$ and $L’$ of the latus rectum are $(a,2a)$ and $(a,-2a)$ respectively. \[\therefore LS=LS’=2a\]

Also, the length of the latus rectum is \[LL’=2a+2a=4a\]

Parabola with its Axis Parallel to the X-axis or Y-axis and Vertex at any Point

Let $(h,k)$ be the vertex of a parabola whose axis is parallel to the x-axis, so that its focus is at $(h+a,k)$, i.e. at a distance $a$ from the focus. The directrix is parallel to y-axis and at a distance $a$ from the vertex i.e. at a distance $(h-a)$ from the y-axis. Hence, equation of the directrix is \[x-h+a=0\]

Parabola with its axis parallel to x-axis or y-axis and vertex at any point

If $(x,y)$ be any point on the parabola, then its distance from the focus $(h+a,k)$ equals its distance from the directrix $x-h+a=0$. \[\therefore [x-(h+a)]^2+(y-k)^2=(x-h+a)^2\] \[\therefore (y-k)^2=4a(x-h)^2\]

This is the equation of the parabola.

Similarly, the equation of a parabola with vertex at $(h,k)$ and axis parallel to the y-axis is \[(x-h)^2=4a(y-k)\]

Special Cases

If the y-coordinates of the vertex and the focus are same, then the axis of the parabola is parallel to x-axis. If the focus lies on the right of the vertex, then the parabola turns to the right and its equation is $(y-k)^2=4a(x-h)$. If the focus lies on the left of the vertex, then the parabola turns to the left and its equation is $(y-k)^2=-4a(x-h)$.

Parabola with axis parallel to x-axis: (y-k)^2=4a(x-h) and (y-k)^2=-4a(x-h)

If the x-coordinates of the vertex and the focus are same, then the axis of the parabola is parallel to y-axis. If the focus lies above the vertex, then the parabola turns upward and its equation is $(x-h)^2=4a(y-k)$. If the focus lies below the vertex, then the parabola turns downward and its equation is $(x-h)^2=-4a(y-k)$.

Parabola with axis parallel to y-axis: (x-h)^2=4a(y-k) and (x-h)^2=-4a(y-k)

General Equation of a Parabola

General Equation of a Parabola

Let $(h,k)$ be the focus and $ax+by+c=0$ be the equation of the directrix of a parabola. Then if $(x,y)$ be any point on the parabola, its equation is \[(x-h)^2+(y-k)^2=\left(\frac{ax+by+c}{\sqrt{a^2+b^2}}\right)^2\]

By solving this equation, we get \[b^2x^2-2abxy+a^2y^2-2x\{(a^2+b^2)h+ac\}\]\[-2y\{(a^2+b^2)k+bc\}+(a^2+b^2)(h^2+k^2)-c^2=0\]

which is in the form of \[(bx-ay)^2+2gx+2fy+k=0\]

Thus, the characteristic property of the equation of a parabola is that the terms of the second degree form a perfect square.



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