I know this is very easy for some of you here , but sadly for me it's not.
So kindly, please help me.
A. How to get the equation of a parabola who opens to the left and whose latus rectum is equal to 8?
B. How to get the equation of a parabola that passes through the point (5,-10)
C. The equation of the parabola whose latus rectum is equal to 10 and opens downward?
The parabola is defined as the set of points, which have the same distance from the focus point and from the directrix line. Therefore the apex will be exactly halfway between the focus and the directrix. The segment of the line parallel to the directrix, which is inside the parabola, is called the latus rectum. Accordingly the size of half of the latus rectum is exactly the same size than the distance from its section point on the parabola to the directrix. Or, stated differently, the quarter of the size of the latus rectum is the distance of that very point to the tangent at the apex.
Therefore for the parabola $y=a\ x^2$ that section point will be located at some point $P(x_0,y_0)$ and you have as additional infos $y_0=a\ x_0^2$ as well as $y_0=x_0/2$. - Equating these and solving for $x_0$ gives solutions $x_{01}=0$, the apex (which is not the searched for solution), respectively $x_{02}=1/2a$. - Thus the latus rectum for the parabola $y=a\ x^2$ does have the size $2\ x_{02}=1/a$. Conversely, when you have given the size of the latus rectum, you could solve for $a$ and get thus the formula of the parabola.
--- rk