# Examples of Parabolas

__Example 1__**:** Find an equation of the parabola having its focus at and as its directrix on the line .

__Solution__**:** Since the focus is on the Y-axis and is also below the directrix, the parabola opens downward, and . Hence the equation of the parabola is . The length of the latus rectum is .

__Example 2__**: **Given the parabola having the equation , 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 . Since , the parabola opens to the right. The focus is at the point .

The equation of the directrix is . The length of the latus rectum is .

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

__Solution__**:** Let be any point of the parabola

Then the length of the latus rectum is , therefore from the above parabola equation:

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