Q.Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola $y^2 = 8x$.

Concept: Standard form of a parabola opening rightward.

The equation $y^2 = 8x$ matches the standard form $y^2 = 4ax$, where the parabola opens to the right with vertex at the origin.

Comparing coefficients: $4a = 8$, so $a = 2$.

For this standard form:

• Focus: $(a, 0) = (2, 0)$
• Axis: the $x$-axis, i.e., $y = 0$
• Directrix: $x = -a$, so $x = -2$
• Latus rectum: the length is $4a = 8$

The latus rectum is the chord through the focus perpendicular to the axis. Its endpoints are $(2, 4)$ and $(2, -4)$, confirming length $8$.

Every parabola has a beautiful geometric definition: it is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). The standard form $y^2 = 4ax$ captures a parabola with vertex at the origin that opens along the positive $x$-axis. The parameter $a$ controls how "wide" the parabola is and directly determines the position of the focus and directrix.

When you see $y^2 = 4ax$, the focus sits at $(a, 0)$, the directrix is the vertical line $x = -a$, and the axis of symmetry is the $x$-axis itself. The latus rectum—the chord through the focus perpendicular to the axis—has length $4a$, which is also the coefficient of $x$ in the equation.

Let's extract these features from $y^2 = 8x$.

1. Identify the standard form and find $a$

The given equation is $y^2 = 8x$. Compare this with the standard form:

Matching coefficients, we have:

This tells us the parabola opens to the right (since the $y^2$ term is isolated and the right-hand side is positive $x$), and the "focal distance" is $2$ units from the vertex.

2. Locate the focus

For the parabola $y^2 = 4ax$, the focus lies on the axis of symmetry at a distance $a$ from the vertex in the direction the parabola opens. Since the vertex is at the origin and the parabola opens rightward:

3. Determine the axis of symmetry

The axis is the line along which the parabola is symmetric. For $y^2 = 4ax$, this is the $x$-axis:

4. Write the equation of the directrix

The directrix is perpendicular to the axis and lies at a distance $a$ from the vertex in the opposite direction to the focus. Since the focus is at $(2, 0)$, the directrix is the vertical line:

5. Find the length of the latus rectum

The latus rectum is the chord through the focus perpendicular to the axis. Its length is always $4a$ for a parabola in standard form:

You can verify this: at $x = 2$ (the focus), substitute into $y^2 = 8x$ to get $y^2 = 16$, so $y = \pm 4$. The distance between $(2, 4)$ and $(2, -4)$ is indeed $8$.