Q.Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $x^2 = -16y$.

Concept: Parabola Standard Form

The given equation $x^2 = -16y$ matches the standard form $x^2 = -4ay$, which opens downward.

Step 1: Identify $a$

Comparing $x^2 = -16y$ with $x^2 = -4ay$, we get $4a = 16$, so $a = 4$.

Step 2: Focus and axis

For $x^2 = -4ay$, the focus is at $(0, -a) = (0, -4)$.

The axis is the $y$-axis (the line $x = 0$).

Step 3: Directrix and latus rectum …

The equation $x^2 = -16y$ is of the form $x^2 = -4ay$, which is the standard form for a parabola that opens downward. In this form, the vertex is at the origin $(0,0)$, the focus lies on the negative y-axis, and the directrix is a horizontal line above the vertex.

Why does this form work? The negative sign tells us the parabola opens opposite to the positive y-direction. The coefficient $4a$ determines the distance from the vertex to the focus (and also to the directrix). Once we identify $a$, everything else follows directly.

Let’s find $a$ by comparing $x^2 = -16y$ with $x^2 = -4ay$.

1. Identify $a$

   We have $-4a = -16$. Dividing both sides by $-4$ gives $a = 4$.

   So the distance from the vertex to the focus is $4$ units, and the same distance from the vertex to the directrix.

2. Focus

   For $x^2 = -4ay$, the focus is at $(0, -a)$. Since $a = 4$, the focus is $(0, -4)$.

3. Axis

   The axis of symmetry is the line through the vertex and the focus. Here, that’s the vertical line $x = 0$, which is the y-axis.

4. Directrix

   The directrix is a horizontal line opposite the focus, at $y = a$. So $y = 4$. …