11.1 Hyperbola Definitions, Standard Equations, and Asymptotes

11.1 Hyperbola Definitions, Standard Equations, and Asymptotes

In Gaokao Mathematics, the hyperbola is one of the essential non-degenerate conic sections. Along with the ellipse and parabola, it forms the core of plane analytic geometry. Mastering the hyperbola requires a rigorous understanding of its first-principle geometric definition, standard algebraic equations, asymptotic behavior, eccentricity, and focal radius properties.

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1. Geometric Definition of the Hyperbola

A hyperbola is defined as the locus of all points $P(x, y)$ in a plane such that the absolute difference of the distances from $P$ to two fixed points $F_1$ and $F_2$ (called the foci) is a constant positive real number $2a$, where $2a$ is strictly less than the distance between the two foci $|F_1F_2| = 2c$.

$$\bigl| |PF_1| - |PF_2| \bigr| = 2a \quad (0 < 2a < 2c)$$

Key Parameters and Boundary Conditions:

• Foci ($F_1, F_2$): Fixed points separated by focal distance $|F_1F_2| = 2c$.
• Focal Distance ($2c$): The distance between the two foci.
• Real Axis Length ($2a$): The constant difference in distance. The segment $A_1A_2$ connecting the vertices is called the real (transverse) axis, of length $2a$.
• Imaginary Axis Length ($2b$): The segment $B_1B_2$ perpendicular to the real axis at the center, of length $2b$, where $b = \sqrt{c^2 - a^2}$.

Critical Gaokao Edge Cases:

1. If $2a = 2c$, the locus degenerates into two opposite rays starting from $F_1$ and $F_2$ along the line $F_1F_2$.
2. If $2a > 2c$, no such point $P$ exists in the real plane (violates the triangle inequality $|PF_1| - |PF_2| \le |F_1F_2|$).
3. If the absolute value sign is omitted, i.e., $|PF_1| - |PF_2| = 2a$, the locus represents only one branch of the hyperbola (the branch wrapping around $F_2$).

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2. Standard Equations and Canonical Forms

Depending on whether the focal axis lies along the $x$-axis or the $y$-axis, the standard equation of a hyperbola centered at the origin $(0,0)$ takes one of two canonical forms.

Distinguishing $a$, $b$, and $c$:

Unlike the ellipse equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$ always holds, in a hyperbola:

• $a$ and $b$ do not have to satisfy $a > b$. We can have $a > b$, $a < b$, or $a = b$.
• $c$ is always the largest parameter: $c > a > 0$ and $c > b > 0$.
• The focal axis orientation is determined solely by the positive term: if the $x^2$ term is positive, the real axis is along the $x$-axis; if the $y^2$ term is positive, the real axis is along the $y$-axis.

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3. Asymptotes: Algebraic Derivation and Geometric Meaning

One of the defining features of a hyperbola is its asymptotes. As a point $P(x, y)$ moves infinitely far along a branch of the hyperbola, the perpendicular distance from $P$ to the asymptote lines approaches zero.

Algebraic Derivation:

For the horizontal hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, we solve for $y$:

$$y^2 = \frac{b^2}{a^2}(x^2 - a^2) \implies y = \pm \frac{b}{a}\sqrt{x^2 - a^2} = \pm \frac{b}{a}|x|\sqrt{1 - \frac{a^2}{x^2}}$$

As $|x| \to \infty$, $\frac{a^2}{x^2} \to 0$, so $y \approx \pm \frac{b}{a}x$. Thus, the lines $y = \pm \frac{b}{a}x$ serve as guidelines that bound the curve.

Quick Asymptote Formula Trick: To quickly obtain the asymptote equations of any hyperbola centered at the origin, set the right-hand constant to $0$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \implies \left(\frac{x}{a} - \frac{y}{b}\right)\left(\frac{x}{a} + \frac{y}{b}\right) = 0 \implies y = \pm \frac{b}{a}x$

Asymptote Family & Asymptote Angle:

• Conjugate Hyperbolas: Hyperbolas of the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = k$ ($k \neq 0$) share the exact same asymptotes $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0$.
• Equilateral (Rectangular) Hyperbola: When $a = b$, the hyperbola equation is $x^2 - y^2 = a^2$ (or $y^2 - x^2 = a^2$). Its asymptotes are $y = \pm x$, which are mutually perpendicular ($90^\circ$). Its eccentricity is always $e = \frac{\sqrt{a^2+a^2}}{a} = \sqrt{2}$.

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4. Eccentricity and Focal Radius Formulas

Eccentricity ($e$):

The ratio $e = \frac{c}{a} > 1$ measures the degree of "opening" or spread of the hyperbola branches.

• As $e \to 1^+$, $\frac{b}{a} = \sqrt{e^2 - 1} \to 0$, meaning the asymptotes are very flat and close to the focal axis.
• As $e \to +\infty$, $\frac{b}{a} \to +\infty$, meaning the asymptotes open widely, becoming nearly vertical.

Focal Radius Formulas:

Let $P(x_0, y_0)$ be a point on the horizontal hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with foci $F_1(-c, 0)$ and $F_2(c, 0)$:

• If $P(x_0, y_0)$ is on the right branch ($x_0 \ge a$): $$|PF_1| = ex_0 + a, \quad |PF_2| = ex_0 - a$$
• If $P(x_0, y_0)$ is on the left branch ($x_0 \le -a$): $$|PF_1| = -ex_0 - a, \quad |PF_2| = -ex_0 + a$$

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5. Worked Gaokao Exam Examples

Example 1: Finding Standard Equation from Asymptotes and Foci

Problem: Find the standard equation of a hyperbola centered at the origin whose foci lie on the $x$-axis, given that its asymptotes are $y = \pm \frac{1}{2}x$ and one of its foci is $F(2\sqrt{5}, 0)$.

Solution:

1. Since the focus is $F(2\sqrt{5}, 0)$, the focal axis is along the $x$-axis and $c = 2\sqrt{5}$, so $c^2 = 20$.
2. The standard equation is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.
3. The asymptote equations are $y = \pm \frac{b}{a}x$. Comparing with $y = \pm \frac{1}{2}x$, we get $\frac{b}{a} = \frac{1}{2} \implies a = 2b$.
4. Using the fundamental identity $c^2 = a^2 + b^2$: $$20 = (2b)^2 + b^2 = 5b^2 \implies b^2 = 4$$
5. Hence $a^2 = 4b^2 = 16$.
6. The standard equation of the hyperbola is $\dfrac{x^2}{16} - \dfrac{y^2}{4} = 1$.

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Example 2: Determining Eccentricity from Asymptote Angle

Problem: A hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0, b>0$) has asymptotes that form an angle of $60^\circ$ with each other. Determine the possible value(s) of its eccentricity $e$.

Solution:

1. The asymptotes are $y = \pm \frac{b}{a}x$. Let $\theta$ be the inclination angle of the asymptote $y = \frac{b}{a}x$ with the positive $x$-axis, so $\tan \theta = \frac{b}{a} > 0$, where $0^\circ < \theta < 90^\circ$.
2. The angle between the two asymptotes is either $2\theta$ (if the angle containing the real axis is $2\theta$) or $180^\circ - 2\theta$.
3. Case 1: The acute angle between asymptotes bisected by the $x$-axis is $60^\circ$, so $2\theta = 60^\circ \implies \theta = 30^\circ$. $$\tan 30^\circ = \frac{b}{a} = \frac{1}{\sqrt{3}} \implies \frac{b^2}{a^2} = \frac{1}{3}$$ $$e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + \frac{1}{3}} = \sqrt{\frac{4}{3}} = \frac{2\sqrt{3}}{3}$$
4. Case 2: The angle between asymptotes containing the $x$-axis is $120^\circ$, so $2\theta = 120^\circ \implies \theta = 60^\circ$. $$\tan 60^\circ = \frac{b}{a} = \sqrt{3} \implies \frac{b^2}{a^2} = 3$$ $$e = \sqrt{1 + 3} = \sqrt{4} = 2$$
5. Thus, the eccentricity $e$ can be either $\dfrac{2\sqrt{3}}{3}$ or $2$.