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 ${\displaystyle f(x)=x^2}$ .

Standard Parabola

The general form of standard parabola is: ${\displaystyle y^2=4ax}$ , where ${\displaystyle a}$ 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. ${\displaystyle y=0}$ line). It means if ${\displaystyle (x,y)}$ lies on parabola. then ${\displaystyle (x,-y)}$ 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 ${\displaystyle O(0,0)}$ i.e. the origin is the vertex of the parabola.
• The fixed point is known as Focus (denoted by ${\displaystyle S}$).
• 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 ${\displaystyle x=a\ldots (1)}$ .

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

Parametric Equations

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

The equation ${\displaystyle x=at^{2}}$ and ${\displaystyle y=2at}$ 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 ${\displaystyle S(h,k)}$, line of directrix as ${\displaystyle ax+by+C=0}$ and point ${\displaystyle P}$ as ${\displaystyle (x_{1},y_{1})\equiv (\alpha ,\beta )}$ whose locus is parabola. As we know that for parabola, ${\displaystyle PS=PN}$ (since e of parabola is 1) ${\displaystyle PS^{2}=PN^{2}}$

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

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

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

See also

• Hyperbola
• Conic section