11.2 Parabola Definitions, Standard Equations, and Focal Properties

11.2 Parabola Definitions, Standard Equations, and Focal Properties

The parabola is unique among conic sections because its eccentricity is precisely equal to $1$ ($e = 1$). In Gaokao Mathematics, parabola problems frequently combine coordinate algebra, focal chord identities, geometric tangency, and optimization.

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1. Geometric Definition of the Parabola

A parabola is the locus of all points $P(x, y)$ in a plane that are equidistant from a fixed point $F$ (the focus) and a fixed straight line $l$ (the directrix) not passing through $F$.

$$|PF| = d(P, l)$$

Fundamental Elements:

• Focal Parameter ($p$): The distance from the focus $F$ to the directrix $l$, where $p > 0$.
• Vertex ($O$): The midpoint of the perpendicular segment from the focus $F$ to the directrix $l$. For standard equations, the vertex is positioned at the origin $(0, 0)$.
• Axis of Symmetry: The line passing through the focus $F$ perpendicular to the directrix $l$.
• Eccentricity ($e$): $e = \frac{|PF|}{d(P, l)} = 1$.

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2. Four Standard Equations and Canonical Orientations

Depending on which axis serves as the axis of symmetry and the direction of opening, the standard equation of a parabola centered at the origin $(0,0)$ takes one of four forms.

Gaokao Memory Tip for Focus & Directrix: The term containing the first power ($x$ or $y$) indicates the axis of symmetry. The coefficient of the first-power term is $\pm 2p$. Therefore, the focus coordinate on that axis is $\frac{\text{coefficient}}{4}$, and the directrix is the negative of that coordinate. Example: For $x^2 = -8y$, the first-power term is $y$ with coefficient $-8$. Focus $y$-coordinate is $\frac{-8}{4} = -2 \implies F(0, -2)$, and directrix is $y = 2$.

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3. Core Algebraic Properties of Focal Chords

Consider the standard parabola $y^2 = 2px$ ($p>0$) with focus $F\left(\frac{p}{2}, 0\right)$. Let a line $AB$ passing through $F$ intersect the parabola at two points $A(x_1, y_1)$ and $B(x_2, y_2)$. The segment $AB$ is called a focal chord.

1. Line Parameterization:

If the line $AB$ has inclination angle $\theta \neq 90^\circ$ (slope $k = \tan \theta$), its equation can be written as $x = my + \frac{p}{2}$ (where $m = \frac{1}{k} = \cot \theta$).

Substituting $x = my + \frac{p}{2}$ into $y^2 = 2px$:

$$y^2 = 2p\left(my + \frac{p}{2}\right) \implies y^2 - 2pmy - p^2 = 0$$

2. Vieta's Relations for Focal Chords:

• Sum of ordinates: $y_1 + y_2 = 2p m$
• Product of ordinates: $y_1 y_2 = -p^2$ (Constant! Independent of line slope $k$)
• Product of abscissas: $x_1 x_2 = \frac{y_1^2 y_2^2}{4p^2} = \frac{(-p^2)^2}{4p^2} = \frac{p^2}{4}$ (Constant!)

3. Focal Chord Length Formulas:

Using the definition $|AF| = x_1 + \frac{p}{2}$ and $|BF| = x_2 + \frac{p}{2}$:

$$|AB| = |AF| + |BF| = x_1 + x_2 + p$$

Expressed in terms of the slope $k$ or inclination angle $\theta$:

$$|AB| = p(m^2 + 1) + p = p\left(\frac{1}{k^2} + 1\right) + p = x_1 + x_2 + p$$ $$|AB| = \frac{2p}{\sin^2 \theta}$$

Minimum Focal Chord Length: Since $\sin^2 \theta \le 1$, the focal chord reaches its minimum length when $\theta = 90^\circ$ ($AB$ is perpendicular to the $x$-axis). The minimum length is $|AB|_{\text{min}} = 2p$, known as the latus rectum.

4. Harmonic Mean Property of Focal Radii:

$$\frac{1}{|AF|} + \frac{1}{|BF|} = \frac{1}{x_1 + \frac{p}{2}} + \frac{1}{x_2 + \frac{p}{2}} = \frac{x_1 + x_2 + p}{\left(x_1 + \frac{p}{2}\right)\left(x_2 + \frac{p}{2}\right)} = \frac{x_1 + x_2 + p}{x_1 x_2 + \frac{p}{2}(x_1 + x_2) + \frac{p^2}{4}}$$

Substituting $x_1 x_2 = \frac{p^2}{4}$:

$$\frac{1}{|AF|} + \frac{1}{|BF|} = \frac{x_1 + x_2 + p}{\frac{p^2}{4} + \frac{p}{2}(x_1 + x_2) + \frac{p^2}{4}} = \frac{x_1 + x_2 + p}{\frac{p}{2}(x_1 + x_2 + p)} = \frac{2}{p}$$

This shows that $\frac{1}{|AF|} + \frac{1}{|BF|} = \frac{2}{p}$ is an invariant constant for any focal chord!

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4. Circle Tangency Theorem on Focal Chords

Theorem: For any focal chord $AB$ of the parabola $y^2 = 2px$, the circle constructed with $AB$ as a diameter is always tangent to the directrix $x = -\frac{p}{2}$.

Proof:

1. Let $M$ be the midpoint of $AB$. Its $x$-coordinate is $x_M = \frac{x_1 + x_2}{2}$.
2. The radius of the circle with diameter $AB$ is $R = \frac{|AB|}{2} = \frac{x_1 + x_2 + p}{2}$.
3. The distance $d$ from the midpoint $M(x_M, y_M)$ to the directrix line $x = -\frac{p}{2}$ is: $$d = x_M - \left(-\frac{p}{2}\right) = \frac{x_1 + x_2}{2} + \frac{p}{2} = \frac{x_1 + x_2 + p}{2}$$
4. Since $d = R$, the distance from the center of the circle to the directrix equals the radius of the circle. Hence, the circle is tangent to the directrix!

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5. Worked Gaokao Exam Examples

Example 1: Finding Parabola Parameter from Focal Chord Length

Problem: A line passing through the focus $F$ of the parabola $y^2 = 2px$ ($p>0$) intersects the parabola at points $A$ and $B$. If the length of chord $AB$ is $8$ and the $x$-coordinate of the midpoint $M$ of $AB$ is $3$, find the value of $p$ and the parabola equation.

Solution:

1. Let $A(x_1, y_1)$ and $B(x_2, y_2)$. The focal chord length formula gives $|AB| = x_1 + x_2 + p = 8$.
2. The $x$-coordinate of the midpoint $M$ is $x_M = \frac{x_1 + x_2}{2} = 3 \implies x_1 + x_2 = 6$.
3. Substitute $x_1 + x_2 = 6$ into the chord length formula: $$6 + p = 8 \implies p = 2$$
4. Therefore, the standard equation of the parabola is $y^2 = 4x$.

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Example 2: Application of Harmonic Focal Chord Identity

Problem: Let $F$ be the focus of the parabola $y^2 = 4x$. A line passing through $F$ with inclination angle $60^\circ$ intersects the parabola at $A$ and $B$. Find $|AB|$ and verify the focal segment reciprocal sum.

Solution:

1. For $y^2 = 4x$, $2p = 4 \implies p = 2$, focus $F(1, 0)$. Directrix $x = -1$.
2. Inclination angle $\theta = 60^\circ \implies \sin 60^\circ = \frac{\sqrt{3}}{2}$.
3. Using the chord length formula $|AB| = \frac{2p}{\sin^2 \theta}$: $$|AB| = \frac{2(2)}{\left(\frac{\sqrt{3}}{2}\right)^2} = \frac{4}{\frac{3}{4}} = \frac{16}{3}$$
4. Verify harmonic mean identity: $\frac{1}{|AF|} + \frac{1}{|BF|} = \frac{2}{p} = \frac{2}{2} = 1$.
5. Both results match Gaokao analytical standards.