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 $x^2 = 6y$.

The parabola $x^2 = 6y$ is in the standard form $x^2 = 4ay$, which opens upward with vertex at the origin.

Comparing coefficients: $4a = 6$, so $a = \frac{3}{2}$.

For this standard form:

• Focus lies at $(0, a) = \left(0, \frac{3}{2}\right)$
• Axis is the line of symmetry: $x = 0$ (the $y$-axis)
• Directrix is the horizontal line $y = -a$, giving $y = -\frac{3}{2}$
• Latus rectum has length $4a = 6$

Why this form tells us everything

A parabola is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix). When the equation is written as $x^2 = 4ay$, the parabola has its vertex at the origin and opens along the $y$-axis. The parameter $a$ encodes the "focal distance"—how far the focus sits from the vertex—and from it we can read off every geometric feature.

The standard form $x^2 = 4ay$ describes a parabola that:

• has vertex at $(0, 0)$,
• opens upward if $a > 0$ (downward if $a < 0$),
• has focus at $(0, a)$,
• has directrix $y = -a$,
• has the $y$-axis as its axis of symmetry,
• has latus rectum (the chord through the focus perpendicular to the axis) of length $4a$.

Our task is to identify $a$ and then extract these features.

Step-by-step extraction

1. Identify the parameter $a$

We have $x^2 = 6y$. Comparing with the standard form $x^2 = 4ay$, we see

Since $a > 0$, the parabola opens upward.

2. Find the focus

The focus of $x^2 = 4ay$ lies at $(0, a)$. Substituting $a = \frac{3}{2}$,

3. Write the equation of the directrix

The directrix is the horizontal line $y = -a$. With $a = \frac{3}{2}$,

4. State the axis of the parabola

The axis of symmetry is the line passing through the vertex and focus, perpendicular to the directrix. Here it is the $y$-axis, or

5. Calculate the length of the latus rectum

The latus rectum is the chord through the focus perpendicular to the axis. Its length is always $4a$. Thus

Summary table