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 $y^2 = -8x$.

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Accepted Answer

> [!TLDR]
> The parabola $y^2 = -8x$ opens leftwards. Its focus is at $(-2, 0)$, axis is the x-axis ($y=0$), directrix is $x = 2$, and the length of the latus rectum is $8$.

This is a standard parabola of the form $y^2 = -4ax$, which opens to the left. The key is to match the given equation to this form and then read off the geometric parameters directly.

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1. **Identify the standard form.**  
   The general equation for a parabola with vertex at the origin and axis along the x-axis is $y^2 = 4ax$ (opens right) or $y^2 = -4ax$ (opens left).  
   Our equation is $y^2 = -8x$. Comparing, we have $-4a = -8$, so $4a = 8$ and therefore $a = 2$.

2. **Focus.**  
   For $y^2 = -4ax$, the focus lies on the negative x-axis at $(-a, 0)$.  
   With $a = 2$, the focus is $(-2, 0)$.

3. **Axis.**  
   The axis is the line of symmetry. For $y^2 = -4ax$, the axis is the x-axis itself, i.e., $y = 0$.

4. **Directrix.**  
   The directrix is a vertical line to the right of the vertex, at $x = a$.  
   So here, $x = 2$.

5. **Length of the latus rectum.**  
   The latus rectum is the chord through the focus perpendicular to the axis. Its length is always $|4a|$ for any parabola of the form $y^2 = \pm 4ax$.  
   Here, $4a = 8$, so the length is $8$.

> [!WARNING]
> A common mistake is to forget the sign. $y^2 = -8x$ means the parabola opens left, so the focus has a *negative* x-coordinate. If you mistakenly treat it as $y^2 = 8x$, you'd get the focus at $(2,0)$ — which is wrong.

> [!TIP]
> You never need to memorise separate formulas for left/right. Just remember: for $y^2 = 4ax$, focus is $(a,0)$, directrix $x = -a$. For $y^2 = -4ax$, just flip the signs: focus $(-a,0)$, directrix $x = a$. The latus rectum length is always $4a$.

> [!ANSWER]
> The focus is $(-2, 0)$, the axis is $y = 0$, the directrix is $x = 2$, and the length of the latus rectum is $8$.

Equation	Opens	Focus	Directrix
$y^2 = 4ax$	right	$(a, 0)$	$x = -a$
$y^2 = -4ax$	left	$(-a, 0)$	$x = a$
$x^2 = 4ay$	up	$(0, a)$	$y = -a$
$x^2 = -4ay$	down	$(0, -a)$	$y = a$

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 $y^2 = -8x$ .

Concept understanding — Parabola Standard Form

Standard Equations of a Parabola

The four standard forms

Where $y^2 = 4ax$ comes from

Latus rectum

Worked example

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 y2=−8xy^2 = -8xy2=−8x.

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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 $y^2 = -8x$ .

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