抛物线pāowùxiàn

Core Concept

A parabola (抛物线) is the set of points equidistant from a fixed point (the focus, 焦点) and a fixed line (the directrix, 准线). That one sentence generates every formula below.

Students often meet the parabola first as the graph of $y = ax^2 + bx + c$. In analytic geometry the emphasis shifts: the same curve is now studied through its focus and directrix, and the standard equations are written differently.

Standard Equations

With the vertex at the origin, there are four orientations. Let $p > 0$:

Equation	Opens	Focus	Directrix
$y^2 = 2px$	right	$\left(\frac{p}{2}, 0\right)$	$x = -\frac{p}{2}$
$y^2 = -2px$	left	$\left(-\frac{p}{2}, 0\right)$	$x = \frac{p}{2}$
$x^2 = 2py$	up	$\left(0, \frac{p}{2}\right)$	$y = -\frac{p}{2}$
$x^2 = -2py$	down	$\left(0, -\frac{p}{2}\right)$	$y = \frac{p}{2}$

Reading the orientation: whichever variable is squared names the axis of symmetry. $y^2 = 2px$ is symmetric about the $x$-axis and opens horizontally; $x^2 = 2py$ is symmetric about the $y$-axis and opens vertically. The sign of the coefficient then decides which of the two directions.

The parameter $p$ is the focal parameter — the distance from the focus to the directrix.

The $\frac{p}{2}$ Rule

Almost every mistake on this topic is a factor-of-two error. Keep this straight:

• The equation carries $2p$.
• The focus sits at distance $\frac{p}{2}$ from the vertex.

So for $y^2 = 8x$: match $2p = 8$, giving $p = 4$, so the focus is at $\left(\frac{4}{2}, 0\right) = (2, 0)$ and the directrix is $x = -2$.

A faster shortcut for the common form $y^2 = 4cx$: the focus is $(c, 0)$ and the directrix is $x = -c$. Since $y^2 = 8x$ is $y^2 = 4(2)x$, we read $c = 2$ immediately.

Key Properties

Property 1: The Focal Distance

For a point $P(x_0, y_0)$ on $y^2 = 2px$, the distance to the focus equals the distance to the directrix:

$|PF| = x_0 + \frac{p}{2}$

This converts a messy distance calculation into simple addition — it is the single most useful parabola formula in exams.

Property 2: Focal Chord Through the Focus

A chord passing through the focus is called a focal chord (焦点弦). If it meets the parabola $y^2 = 2px$ at $A(x_1, y_1)$ and $B(x_2, y_2)$:

$|AB| = x_1 + x_2 + p$

Property 3: The Latus Rectum

The focal chord perpendicular to the axis is the latus rectum (通径), with length $2p$. It is the shortest possible focal chord.

Property 4: Eccentricity

Every parabola has eccentricity exactly $e = 1$. This is what separates it from the ellipse ($e < 1$) and hyperbola ($e > 1$).

Real-World Applications

Parabolic reflectors exploit a geometric fact: rays travelling parallel to the axis all reflect through the focus. This is why satellite dishes, car headlights, and solar concentrators are parabolic in cross-section.

CSCA Practice Problems

💡 Note: These problems follow the CSCA exam syllabus and Chinese standardized test formats.

Example 1: Basic (Difficulty ★★☆☆☆)

The focus of the parabola $y^2 = 12x$ is:

Options:

• A. $(3, 0)$
• B. $(6, 0)$
• C. $(0, 3)$
• D. $(12, 0)$

Detailed Solution:

1. Since $y$ is squared, the parabola opens horizontally with its axis along the $x$-axis. Options C is eliminated immediately.

2. Match to $y^2 = 2px$: $2p = 12$, so $p = 6$.

3. Focus at $\left(\frac{p}{2}, 0\right) = (3, 0)$.

Option B is the trap for anyone who stops at $p = 6$ and forgets to halve.

Answer: A

---

Example 2: Intermediate (Difficulty ★★★☆☆)

Point $P$ lies on the parabola $y^2 = 4x$, and its distance to the focus is $5$. Find the $x$-coordinate of $P$.

Options:

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Detailed Solution:

1. Match to $y^2 = 2px$: $2p = 4$, so $p = 2$ and the focus is at $(1, 0)$, directrix $x = -1$.

2. Apply the focal-distance property rather than the distance formula: $|PF| = x_0 + \frac{p}{2} = x_0 + 1$

3. Set equal to the given distance: $x_0 + 1 = 5 \implies x_0 = 4$

4. Sanity check: at $x_0 = 4$, $y^2 = 16$, so $y = \pm 4$. Distance from $(4, 4)$ to $(1,0)$ is $\sqrt{9 + 16} = 5$. ✓

Answer: B

Why this shortcut matters: the direct route requires solving $\sqrt{(x_0-1)^2 + y_0^2} = 5$ together with $y_0^2 = 4x_0$ — doable, but three times as long and far more error-prone.

Common Mistakes

❌ Mistake 1: Confusing $p$ and $\frac{p}{2}$

The equation contains $2p$; the focus is at $\frac{p}{2}$. For $y^2 = 2px$ these differ by a factor of four, so this error changes the answer dramatically.

❌ Mistake 2: Getting the orientation backwards

$y^2 = 2px$ opens horizontally (along $x$), even though $y$ is the squared variable. Remember: the squared variable names the axis of symmetry, and the curve opens along the other variable's direction.

❌ Mistake 3: Using the distance formula when the focal-distance property applies

If a question mentions "distance from a point on the parabola to the focus", reach for $|PF| = x_0 + \frac{p}{2}$ first.

❌ Mistake 4: Assuming the vertex is at the origin

Equations like $(y-1)^2 = 4(x-2)$ describe a parabola with vertex $(2, 1)$. Shift the focus and directrix by the same amount.

Study Tips

1. ✅ Memorise the four standard forms as a table, not as four separate facts
2. ✅ Write out $2p = \text{coefficient}$ explicitly before computing the focus
3. ✅ Default to the focal-distance property for any focus-related distance
4. ✅ Sketch the curve — orientation errors become obvious immediately
5. ✅ Compare all three conics side by side ($e = 1$ for parabola) to keep them distinct

---

💡 Exam Tip: 根据抛物线方程求焦点或准线 is among the top-five CSCA analytic-geometry knowledge points. It is nearly always a single-step question — so the marks come down to getting the $\frac{p}{2}$ right.