Examples of Parabola

Example 1: Find an equation of the parabola having its focus at \left( {0, - 3} \right) and as its directrix 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 an 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 directrix, and the length of the latus rectum.
Solution: Compare 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 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.

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