Q.Find the equation of the parabola which is symmetric about the $y$-axis, and passes through the point $(2, -3)$.

Concept: Standard form of a parabola symmetric about the $y$-axis.

A parabola symmetric about the $y$-axis has its vertex at the origin and opens either upward or downward. Its equation takes the form:

where $a$ is a constant that determines the direction and "width" of the parabola.

Since the parabola passes through $(2, -3)$, substitute $x = 2$ and $y = -3$:

Substituting back into the standard form: …

When a parabola is symmetric about the $y$-axis, the axis itself is the parabola's axis of symmetry. This means the vertex sits on the $y$-axis and the parabola opens either upward or downward. The standard form for such a parabola with vertex at the origin is $x^2 = 4ay$, where $a$ is a parameter that controls both the "width" and the direction of opening.

Why this form? Because squaring $x$ ensures that points $(x, y)$ and $(-x, y)$ both satisfy the equation — perfect symmetry about the $y$-axis. The parameter $a$ determines the focus location: if $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

Since we're given a point the parabola passes through, we can substitute it directly to find $a$.

1. Write the general equation.

   For a parabola symmetric about the $y$-axis with vertex at the origin:

2. Substitute the given point $(2, -3)$.

   The parabola passes through $(2, -3)$, so $x = 2$ and $y = -3$ must satisfy the equation:

3. Solve for $a$. …