Question

Focal diameter 8 and focus on the negative y-axis.

Short answer

The equation of parabola having a vertex at the origin, focal diameter is 8 and focus is on the negative x -axis is $x^2=-8y$ .

Step-by-step solution

Step 1. Given information.

The parabola vertex is at the origin, focal diameter is 8 and focus is on the negative y-axis.

Step 2. The equation of a parabola that satisfies the conditions.

The focus is on the negative y-axis, so the parabola opens downwards which means p<0.

On comparing, focal diameter with 4p, we get the value for p but with a negative sign, as the parabola opens downwards.

On substituting the value of p in the equation $x^2=4py$ to obtain an equation of a parabola

The vertex at the origin i.e.,$(0,0)$.
The focal diameter is 8.

The focal diameter of a parabola $x^2=4py$ is $4p$.

Then, we get –

$\begin{array}{l}4p=8\\ \Rightarrow p=2\end{array}$

(Dividing by 2 both the side)
The focus is on the negativey-axis.
The focus is $\text{(0,-2)}$ and $\text{p=-2}$.

Now, substituting the value of in equation$x^2=4py$, we get the equation as –

$\begin{array}{l}{x}^2=4(-2)y\\ \Rightarrow x^2=-8y.\end{array}$