# 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 $$P$$.