# 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 $P$.