Question

Directrix has y-intercept 6.

Short answer

The equation of parabola having a vertex at the origin and directrix has y -6 -intercept $x^2=-24y$.

Step-by-step solution

Step 1.Given information.

The parabola vertex is at the origin and directrix y intercept 6 .

Step 2. The parabola vertex is at the origin and directrix have  intercept 6 .

The directrix y -intercept is positive so, it means the focus should be negative, the parabola opens downwards and the coordinates of focus are obtained.

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 directrix of parabola has y-intercept 6 , i.e., $y=6$.

So, the coordinate of focus will be$(0,-6)$and$p=-6$, it means that the parabola opens downwards.

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

$\begin{array}{l}{x}^2=4(-6)y\\ \Rightarrow x^2=-24y.\end{array}$