Hyperbola

For any point $P(x_1, y_1)$ lying on the right-hand branch ($x_1 \ge a$) of the standard hyperbola, its focal distances track as: \begin{formulabox}[Focal Distances] \[ SP = ex_1 - a, \qquad S'P = ex_1 + a \] \[ S'P - SP = 2a \quad (\text{constant value equal to transverse axis length}) \] \end{formulabox}

[leftmargin=*]
• Both the primary hyperbola and its conjugate orientation share the exact same pair of asymptotic lines.
• Eccentricity Reciprocal Identity: If $e_1$ represents the eccentricity of the primary curve and $e_2$ is the eccentricity of its conjugate curve, they satisfy the balanced identity: \[ \frac{1}{e_1^2} + \frac{1}{e_2^2} = 1. \]
• The conjugate hyperbola layout has its transverse axis of length $2b$ oriented vertically along the $y$-axis, with its foci situated at $(0, \pm be)$.

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\begin{formulabox}[Asymptotes of $\ds\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$] An asymptote is a straight line that passes through the center of the hyperbola and stands tangent to the curve branches at infinity. Their linear equations are: \[ y = \frac{b}{a}x \quad \text{and} \quad y = -\frac{b}{a}x \] Combined joint equation: $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \implies b^2x^2 - a^2y^2 = 0$. \end{formulabox}

1. The asymptotes always intersect at the geometric center of the curve.
2. Constant Separation Law: Hyperbola Equation $-$ Joint Asymptotes Equation $=$ Constant. The difference between $\ds\frac{x^2}{a^2}-\frac{y^2}{b^2}-1=0$ and $\ds\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$ is strictly equal to $1$.
3. The total interior angle formed between the two asymptotes evaluates to: $2\tan^{-1}\left(\ds\frac{b}{a}\right)$.
4. If a tangent line drawn at any point $P$ on the hyperbola cuts the asymptotes at points $A$ and $B$, the segment $AB$ is bisected exactly at the point of contact $P$.

\begin{importantbox}[Relation between Hyperbola, Asymptotes and Conjugate] The joint equations satisfy a perfect algebraic distribution balance: \[ \text{Hyperbola Formula} + \text{Conjugate Formula} = 2 \times \text{Joint Asymptotes Formula} \] \[ \left(\frac{x^2}{a^2}-\frac{y^2}{b^2}-1\right) + \left(\frac{y^2}{b^2}-\frac{x^2}{a^2}-1\right) = 2 \cdot \left(\frac{x^2}{a^2}-\frac{y^2}{b^2}\right) - 2. \] \end{importantbox}

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\begin{formulabox}[Parametric Equations] The coordinates of any variable point on the standard hyperbola $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ can be expressed in terms of an eccentric angle parameter $\theta$ as: \[ x = a\sec\theta, \quad y = b\tan\theta, \quad \theta \in \R \setminus \left\{(2k+1)\frac{\pi}{2}, \, k \in \mathbb{Z}\right\} \] Alternatively, it can be parameterized using hyperbolic functions as: \[ x = a\cosh t, \quad y = b\sinh t, \quad t \in \mathbb{R}. \] \end{formulabox}

\begin{formulabox}[Tangent: Point and Parametric Form] At a given point $(x_1, y_1)$ on the curve: $x y_1 + y x_1 = 2c^2$

At a given parameter point $t$: $x + t^2 y = 2ct$ \end{formulabox}

\begin{formulabox}[Normal] Differentiating the parametric equations yields the normal line equation at parameter $t$: \[ t^3 x - ty = c(t^4 - 1) \] \end{formulabox}

The straight line chord connecting two parameter points $t_1$ and $t_2$ is modeled by: \[ x + t_1 t_2 y = c(t_1 + t_2). \]

1. Normals Concurrency Condition: The four normals drawn at parameters $t_1, t_2, t_3,$ and $t_4$ are concurrent if and only if: $t_1 t_2 t_3 t_4 = 1$.
2. Re-intersection Rule: If the normal line drawn at parameter $t$ intersects the rectangular hyperbola again at a new parameter point $t'$, they satisfy the relationship: $t' = -\frac{1}{t^3}$.
3. Cyclic Inscription Property: If a circle cuts the rectangular hyperbola $xy = c^2$ at four points $t_1, t_2, t_3,$ and $t_4$, the parameters satisfy: $t_1 t_2 t_3 t_4 = 1$.

\begin{formulabox}[Point Form ($T = 0$)] \[ \frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1 \] \end{formulabox}

\begin{formulabox}[Parametric Form at $\theta$] \[ \frac{x\sec\theta}{a} - \frac{y\tan\theta}{b} = 1 \] \end{formulabox}

\begin{formulabox}[Slope Form — slope $m$] \[ y = mx \pm \sqrt{a^2 m^2 - b^2} \] \end{formulabox}

\begin{importantbox}[Condition of Tangency] The straight line $y = mx + c$ is tangent to the standard hyperbola $\ds\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ if and only if: \[ c^2 = a^2 m^2 - b^2 \implies c = \pm \sqrt{a^2m^2-b^2}. \] Note: For real tangent lines to exist, the expression under the radical must be strictly positive ($a^2m^2 - b^2 > 0$), which restricts the allowed slopes to: $|m| > \ds\frac{b}{a}$. \end{importantbox}

\begin{formulabox}[Point Form] \[ \frac{a^2 x}{x_1} + \frac{b^2 y}{y_1} = a^2 + b^2 \] \end{formulabox}

\begin{formulabox}[Parametric Form at $\theta$] \[ ax\cos\theta + by\cot\theta = a^2 + b^2 \] \end{formulabox}

\begin{formulabox}[Slope Form — slope $m$] \[ y = mx \mp \frac{(a^2+b^2)m}{\sqrt{a^2 - b^2 m^2}} \] \end{formulabox}

Similar to the ellipse configuration, a maximum of four real normals can be drawn from an external coordinate point to a hyperbola.

The straight line connecting the two points of tangency where tangent lines drawn from an external point $P(x_1, y_1)$ touch the hyperbola is the chord of contact: \begin{formulabox}[Chord of Contact from $(x_1, y_1)$: $T = 0$] \[ \frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1 \] \end{formulabox}

If a chord line is bisected exactly at an interior point $M(h, k)$, its equation can be found using the $T = S_1$ formula: \begin{formulabox}[Midpoint Form: $T = S_1$] \[ \frac{hx}{a^2} - \frac{ky}{b^2} = \frac{h^2}{a^2} - \frac{k^2}{b^2} \] The corresponding slope of this bisected chord evaluates to: $m = \ds\frac{b^2 h}{a^2 k}$. \end{formulabox}

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\begin{formulabox}[Polar of $(x_1, y_1)$ w.r.t. Hyperbola] \[ \frac{x x_1}{a^2} - \frac{y y_1}{b^2} = 1 \quad \Longleftrightarrow \quad T = 0 \] The point $(x_1, y_1)$ is defined as the pole, and the resulting line is its polar line. \end{formulabox}

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\begin{formulabox}[Director Circle of Hyperbola] The locus of the point of intersection of two tangents to a hyperbola that meet at a constant angle of $90^\circ$ is a concentric circle called the director circle. Its equation is: \[ x^2 + y^2 = a^2 - b^2 \] \end{formulabox}

[leftmargin=*]
• This circle exists as a real curve only when the semi-major axis is strictly greater than the semi-minor axis ($a > b$).
• When $a = b$ (rectangular hyperbola), the radius vanishes ($a^2 - b^2 = 0$), causing the director circle to collapse into a single point at the center $(0,0)$. This indicates that the only perpendicular tangents that can be drawn to a rectangular hyperbola are its asymptotes, which intersect at the center.
• When $a < b$, no real points exist in the plane from which perpendicular tangents can be drawn to the hyperbola.

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\begin{importantbox}[Reflection Property] An incoming ray of light directed toward one focus $S$ of a hyperbola reflects off the outer curve boundary surface and is diverted along a line that appears to originate from the other focus $S'$.

The normal line at any point $P$ on a hyperbola acts as the internal angle bisector of the focal angle $\angle SPS'$. \end{importantbox}

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1. To find their points of intersection, solve the two quadratic equations simultaneously by eliminating one variable to generate a quartic polynomial.
2. To find their common tangents, express the tangent line in slope form for both conics independently, equate their intercept components, and solve for the common slope parameter $m$.
3. The linear combination family equation $S_1 + \lambda S_2 = 0$ represents the system of conics passing through the common intersection points. If a value of $\lambda$ eliminates the quadratic terms, the equation simplifies to their common chord line equation $S_1 - S_2 = 0$.

\begin{importantbox}[Common Chord of Two Conics] If $S_1 = 0$ and $S_2 = 0$ are two normalized quadratic conics, then the linear difference equation $S_1 - S_2 = 0$ defines the equation of their common chord line segment. \end{importantbox}

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1. Asymptotic Area Property: The total area of the bounding triangle formed by drawing a variable tangent line to a hyperbola and its two asymptotes is constant and always equal to $ab$.
2. If a circle intersects the rectangular hyperbola $xy = c^2$ at four points $t_1, t_2, t_3,$ and $t_4$, the product of their parameters is always equal to unity: $t_1 t_2 t_3 t_4 = 1$.
3. The locus of the foot of the perpendicular dropped from either focus to any variable tangent line of a hyperbola is its auxiliary circle $x^2 + y^2 = a^2$.
4. The segment of any variable tangent line intercepted between the two asymptotes is always bisected exactly at its point of contact $P$ with the hyperbola branch.

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1. Prove that the product of the lengths of the perpendicular distances dropped from the two foci $S$ and $S'$ to any variable tangent line of the standard hyperbola $\ds\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is constant and always equal to $b^2$.
2. Find the equation of the chord of the rectangular hyperbola $xy = c^2$ that is bisected exactly at an interior point $M(h, k)$.
3. Show that the total area of the triangle formed by any variable tangent line to a hyperbola with its two asymptotes is constant.
4. Find the locus of the midpoints of the chords of the hyperbola $\ds\frac{x^2}{9}-\frac{y^2}{4}=1$ that are drawn parallel to the line $y = 2x$.
5. If $e_1$ and $e_2$ represent the eccentricities of a primary standard hyperbola and its corresponding conjugate hyperbola, prove that $\ds\frac{1}{e_1^2}+\frac{1}{e_2^2}=1$.

1. Find the equation of the chord of the rectangular hyperbola $xy = c^2$ that is bisected exactly at the point $(h, k)$:
   1. $x k + y h = 2hk$
   2. $x h + y k = 2hk$
   3. $x k + y h = hk$
   4. $x + t^2y = 2ct$
2. Find the locus of the midpoints of the chords of $\ds\frac{x^2}{9} - \frac{y^2}{4} = 1$ that are parallel to the line $y = 2x$ (derived using the midpoint chord slope formula):
   1. $2x - 9y = 0$
   2. $4x - 9y = 0$
   3. $9x - 2y = 0$
   4. $x - 2y = 0$
3. If the product of the perpendicular distances from the foci to a tangent line of the hyperbola $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $16$, find the length of the conjugate axis ($2b$):
   1. $8$
   2. $4$
   3. $16$
   4. $2$
4. The area of the triangle formed by any variable tangent line to the hyperbola $\ds\frac{x^2}{16} - \frac{y^2}{9} = 1$ with its asymptotes is:
   1. $12$
   2. $24$
   3. $6$
   4. $144$
5. If the eccentricity of a primary hyperbola is $e_1 = \sqrt{3}$, find the eccentricity $e_2$ of its matching conjugate hyperbola:
   1. $\sqrt{1.5}$
   2. $\sqrt{3}/2$
   3. $2$
   4. $\sqrt{2}$
   \Answer Key: 1. (A) 2. (B) 3. (A) 4. (A) 5. (A)

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\begin{formulabox}[All Hyperbola Formulas at a Glance] Standard Equation Parameters: $\ds\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$; $b^2=a^2(e^2-1)$; $e>1$.

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