Examples of Parabolas

Example 1: Find an equation of the parabola having its focus at \left( {0, - 3} \right) and as its directrix on the line y = 3.

Solution: Since the focus is on the Y-axis and is also below the directrix, the parabola opens downward, and a = - 3. Hence the equation of the parabola is {x^2} = - 12y. The length of the latus rectum is |4\left( { - 3} \right)| = 12.

Example 2: Given the parabola having the equation {y^2} = 7x, find the coordinates of the focus, the equation of the directrix, and the length of the latus rectum.

Solution: Compared with the general equation, here we have 4a = 7 \Rightarrow a = \frac{7}{4}. Since a > 0, the parabola opens to the right. The focus is at the point F\left( {\frac{7}{4},0} \right).
The equation of the directrix is x = - \frac{7}{4}. The length of the latus rectum is 7.

Example 3: Show that the ordinate at any point P of the parabola is a mean proportional between the length of the latus rectum and the abscissa of P.

Solution: Let P\left( {x,y} \right) be any point of the parabola
{y^2} = 4ax

Then the length of the latus rectum is l = 4a, therefore from the above parabola equation:

\begin{gathered} 4ax = {y^2} \\ \Rightarrow \left( {Length\,oflatus\,rectum} \right)\left( {abscissa\,of\,P} \right) = {\left( {ordinate\,of\,P} \right)^2} \\ \end{gathered}

This shows that the ordinate at any point P of the parabola is a mean proportional between the length of the latus rectum and the abscissa of P.