Parabola

Standard form of parabola

A parabola is the locus of a point which moves in a plane such that its distance from a fixed point (i.e. focus) is always equal to its distance from a fixed straight line (directrix). A parabola is a graph of a quadratic function, such as $f(x)=x^2$ .

Standard Parabola

The general form of standard parabola is: $y^2=4ax$ , where $a$ is a constant.

Important terms

• The straight line passing through the focus and perpendicular to the directrix is called the Axis of the Parabola. The parabola is a symmetrical about its axis (i.e. $y=0$ line). It means if $(x,y)$ lies on parabola. then $(x,-y)$ also lies on other sides of the axis.
• The point which bisects every chord of the conic passing through it is called the Centre of the parabola.
• The points of intersection of the conic section and the axis called Vertex. Vertex is point $O(0,0)$ i.e. the origin is the vertex of the parabola.
• The fixed point is known as Focus (denoted by $S$).
• The fixed line is known as Directrix.
• A chord passing through the focus is known as focal chord.
• The straight line through focus and perpendicular to directrix is known as Axis.
• The focal chord which is perpendicular to the axis is known as Latus Rectum. Since it passes through the focus of parabola, the equation of latus rectum is $x=a\ldots (1)$ .

Solving equation (1) and $y^2=4ax$ we get: $y=\pm 2a$ .

Parametric Equations

From the equation of parabola, we can write ${\frac {y}{2a}}={\frac {2x}{y}}=t$ where $t$ is a parameter. Then, $y=2at$ and $x=at^{2}$

The equation $x=at^{2}$ and $y=2at$ are called parametric equations.

General Form of Parabola

Finding the equation of parabola when focus and line of directrix are give Assume that the focus is $S(h,k)$, line of directrix as $ax+by+C=0$ and point $P$ as $(x_{1},y_{1})\equiv (\alpha ,\beta )$ whose locus is parabola. As we know that for parabola, $PS=PN$ (since e of parabola is 1) $PS^{2}=PN^{2}$

$\Rightarrow (x_{1}-h)^{2}+(y_{1}-k)^{2}=\left({\frac {(ax_{1}+by_{1}+C)^{2}}{\sqrt {a^{2}+b^{2}}}}\right)^{2}$

After simpligying the above equation and then replacing $x_1$ by $x$ and $y_1$ by $y$ , we get the required equation to parabola. The simplified form of general equation of parabola would look like:

$(bx-ax)^{2}+2gx+2fy+c=0$, Where g, f and c are real constants.

See also

• Hyperbola
• Conic section