\displaystyle \textbf{1. Definition of a Parabola}
\displaystyle \text{A parabola is the locus of a point which is equidistant from a fixed point (called focus)}
\displaystyle \text{and a fixed line (called directrix).}
\displaystyle \text{Thus, if }(\alpha,\beta)\text{ is the focus and }ax+by+c=0\text{ is the equation of the directrix, then}
\displaystyle \text{its equation is}
\displaystyle (x-\alpha)^2+(y-\beta)^2=\frac{(ax+by+c)^2}{a^2+b^2}
\displaystyle \text{This equation is of the form}
\displaystyle ax^2+2hxy+by^2+2gx+2fy+c=0
\displaystyle \text{satisfying the conditions}
\displaystyle abc+2fgh-af^2-bg^2-ch^2\neq0\quad\text{and}\quad h^2-ab=0
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\displaystyle \textbf{2. Axis of a Parabola}
\displaystyle \text{The straight line passing through the focus and perpendicular to the directrix}
\displaystyle \text{is called the axis of the parabola.}
\\

\displaystyle \textbf{3. Vertex of a Parabola}
\displaystyle \text{The point of intersection of the parabola and its axis is called the vertex}
\displaystyle \text{of the parabola.}
\\

\displaystyle \textbf{4. Latus-rectum}
\displaystyle \text{A chord passing through the focus and perpendicular to the axis is called the}
\displaystyle \text{latus-rectum.}
\\

\displaystyle \textbf{5. Focal Chord}
\displaystyle \text{Any chord passing through the focus of a parabola is called its focal chord.}
\\

\displaystyle \textbf{6. Double Ordinate}
\displaystyle \text{Any chord perpendicular to the axis of a parabola is called a double ordinate.}
\\

\displaystyle \textbf{7. Standard Forms of a Parabola}
\displaystyle \begin{array}{|l|c|c|c|c|}\hline & y^2=4ax & y^2=-4ax & x^2=4ay & x^2=-4ay\\ \hline \text{Coordinates of vertex} & (0,0) & (0,0) & (0,0) & (0,0)\\ \hline \text{Coordinates of focus} & (a,0) & (-a,0) & (0,a) & (0,-a)\\ \hline \text{Equation of the directrix} & x=-a & x=a & y=-a & y=a\\ \hline \text{Equation of the axis} & y=0 & y=0 & x=0 & x=0\\ \hline \text{Length of the latus\text{-}rectum} & 4a & 4a & 4a & 4a\\ \hline \text{Focal distance of a point }P(x,y) & a+x & a-x & a+y & a-y\\ \hline \end{array}
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