In-Depth Study of the Ellipse

1. The Focus-Directrix Definition ($e < 1$)

An ellipse is the planar locus of a point $P(x, y)$ that moves such that the ratio of its distance from a fixed focus $S(ae, 0)$ to its distance from a fixed directrix line $D: x = a/e$ is a constant eccentricity $e \in (0, 1)$: $$\frac{SP}{PM} = e \iff SP = e \cdot PM$$ Let $P(x, y)$ be any point on the curve. Then: $$SP^2 = e^2 PM^2 \implies (x - ae)^2 + y^2 = e^2 \left(x - \frac{a}{e}\right)^2 = (ex - a)^2$$ $$x^2 - 2aex + a^2 e^2 + y^2 = e^2 x^2 - 2aex + a^2$$ Canceling $-2aex$ and grouping terms: $$(1 - e^2)x^2 + y^2 = a^2(1 - e^2) \iff \frac{x^2}{a^2} + \frac{y^2}{a^2(1 - e^2)} = 1$$ Defining the minor semi-axis $b > 0$ by the fundamental relation: $$\mathbf{b^2 \equiv a^2(1 - e^2) \iff e = \sqrt{1 - \frac{b^2}{a^2}} < 1}$$ yields the universal canonical equation of the ellipse: $$\mathbf{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0)}$$

2. The Two Foci and The Constant Sum of Focal Radii

By symmetry about the $y$-axis, the ellipse possesses a second focus $S'(-ae, 0)$ and a second directrix $D': x = -a/e$. The focal distances to any point $P(x, y)$ on the ellipse are: $$SP = a - ex, \qquad S'P = a + ex$$ Adding the two distances: $$SP + S'P = (a - ex) + (a + ex) = \mathbf{2a = \text{Constant Everywhere!}}$$ The Gardener's / Focal Distance Theorem: An ellipse is the locus of all points whose sum of distances from two fixed foci $S$ and $S'$ is constant and equal to the major axis $2a$.

3. Canonical Geometric Elements

• Center ($C$): Origin $(0, 0)$.
• Major Axis: Segment $A'A$ along the $x$-axis, length $2a$.
• Minor Axis: Segment $B'B$ along the $y$-axis, length $2b$.
• Foci: $S(ae, 0)$ and $S'(-ae, 0)$, distance between foci $SS' = 2ae$.
• Directrices: $x = \pm a/e$, distance between directrices $2a/e$.
• Latus Rectum: Chord through focus perpendicular to major axis. Substituting $x = ae$: $$\frac{a^2 e^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \frac{y^2}{b^2} = 1 - e^2 = \frac{b^2}{a^2} \implies y = \pm \frac{b^2}{a} \implies \mathbf{Length = \frac{2b^2}{a}}$$

1. The Auxiliary Circle

The circle described on the major axis $A'A$ of the ellipse as diameter is termed the auxiliary circle. Its equation is: $$x^2 + y^2 = a^2$$ Let $P(x, y)$ be any point on the ellipse. Draw a vertical ordinate through $P$ and extend it to meet the auxiliary circle at $Q(x, Y)$. Because $Q$ lies on the auxiliary circle, $x = a \cos \phi$, where $\phi$ is the angle $\angle OCQ$ measured from the major axis. Substituting $x = a \cos \phi$ into the ellipse equation: $$\frac{a^2 \cos^2 \phi}{a^2} + \frac{y^2}{b^2} = 1 \implies \cos^2 \phi + \frac{y^2}{b^2} = 1 \implies \frac{y^2}{b^2} = \sin^2 \phi \implies y = b \sin \phi$$ The angle $\phi \in [0, 2\pi)$ is termed the eccentric angle of point $P$.

2. Standard Parametric Form

The parametric coordinates of any point on the ellipse are: $$\mathbf{P(\phi) = (a \cos \phi, \; b \sin \phi)}$$ Notice the vertical scaling relation between the ellipse and its auxiliary circle: $$\frac{y_P}{Y_Q} = \frac{b \sin \phi}{a \sin \phi} = \frac{b}{a} = \text{Constant}$$ An ellipse is an auxiliary circle uniformly compressed vertically by the factor $b/a$! Consequently, the area of the ellipse is: $$\operatorname{Area}(\text{Ellipse}) = \frac{b}{a} \times \operatorname{Area}(\text{Auxiliary Circle}) = \frac{b}{a}(\pi a^2) = \mathbf{\pi a b}$$

1. Equations of the Tangent

1. Point Form: Tangent at $P(x_1, y_1)$ on the ellipse: $$\mathbf{\frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1}$$
2. Parametric Form: Tangent at $P(\phi) = (a\cos\phi, b\sin\phi)$: $$\mathbf{\frac{x \cos \phi}{a} + \frac{y \sin \phi}{b} = 1}$$
3. Slope Form: A line $y = mx + c$ is tangent to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ if and only if $c^2 = a^2 m^2 + b^2$: $$\mathbf{y = m x \pm \sqrt{a^2 m^2 + b^2}}$$

2. The Director Circle of the Ellipse

Let $P(h, k)$ be the point of intersection of two mutually perpendicular tangents to the ellipse. The slope form of a tangent passing through $(h, k)$ is: $$k - mh = \pm \sqrt{a^2 m^2 + b^2} \iff (k - mh)^2 = a^2 m^2 + b^2$$ Expanding and grouping as a quadratic in the slope $m$: $$(h^2 - a^2)m^2 - 2hk m + (k^2 - b^2) = 0$$ If the two tangents are perpendicular, the product of their slopes must be $-1$: $$m_1 m_2 = \frac{k^2 - b^2}{h^2 - a^2} = -1 \iff k^2 - b^2 = -(h^2 - a^2) \iff h^2 + k^2 = a^2 + b^2$$ Replacing $(h, k)$ with current coordinates $(x, y)$: $$\mathbf{x^2 + y^2 = a^2 + b^2}$$ The Director Circle Theorem: The locus of the point of intersection of two mutually perpendicular tangents to an ellipse is a concentric circle of radius $R = \sqrt{a^2 + b^2}$, termed the director circle!

3. Equation of the Normal

The normal line at $P(x_1, y_1)$ is perpendicular to the tangent: $$\mathbf{\frac{a^2 x}{x_1} - \frac{b^2 y}{y_1} = a^2 - b^2}$$ In parametric coordinates $P(\phi)$: $$\mathbf{a x \sec \phi - b y \csc \phi = a^2 - b^2}$$

1. Definition of Conjugate Diameters

A diameter of an ellipse is a chord passing through its center $C(0, 0)$. Two diameters $y = m_1 x$ and $y = m_2 x$ are said to be conjugate diameters if each bisects all chords parallel to the other. The algebraic condition for conjugacy is: $$\mathbf{m_1 m_2 = -\frac{b^2}{a^2}}$$ In parametric terms, if $CP$ is a semi-diameter with endpoint $P(\phi) = (a\cos\phi, b\sin\phi)$, its conjugate semi-diameter $CD$ has endpoint $D$ whose eccentric angle differs by $\pi/2$: $$\phi_D = \phi + \frac{\pi}{2} \implies \mathbf{D = (-a \sin \phi, \; b \cos \phi)}$$

2. Apollonius' First Theorem (Sum of Squared Semi-Diameters)

The sum of the squares of any two conjugate semi-diameters is constant and equal to the sum of the squares of the semi-axes: $$CP^2 = a^2 \cos^2 \phi + b^2 \sin^2 \phi$$ $$CD^2 = a^2 \sin^2 \phi + b^2 \cos^2 \phi$$ Adding the two equations: $$CP^2 + CD^2 = a^2(\cos^2 \phi + \sin^2 \phi) + b^2(\sin^2 \phi + \cos^2 \phi) = \mathbf{a^2 + b^2 = \text{Constant!}}$$

3. Apollonius' Second Theorem (Area of Circumscribing Parallelogram)

The area of the parallelogram formed by the tangents drawn at the extremities of any pair of conjugate diameters is constant and equal to the area of the rectangle formed by the principal axes: $$\mathcal{A} = 4 \left| x_P y_D - x_D y_P \right| = 4 |(a\cos\phi)(b\cos\phi) - (-a\sin\phi)(b\sin\phi)| = 4 a b(\cos^2 \phi + \sin^2 \phi) = \mathbf{4 a b}$$

1. The Optical/Acoustic Reflection Property

Let $P(x_1, y_1)$ be any point on the ellipse with foci $S(ae, 0)$ and $S'(-ae, 0)$. Let the normal at $P$ meet the major axis at $G$. By the properties of the normal: $$CG = e^2 x_1 \implies SG = ae - e^2 x_1 = e(a - ex_1) = e \cdot SP, \quad S'G = ae + e^2 x_1 = e(a + ex_1) = e \cdot S'P$$ Therefore: $$\frac{SG}{S'G} = \frac{SP}{S'P}$$ By the angle bisector theorem, the normal $PG$ is the internal bisector of the focal angle $\angle SPS'$! Consequently, the tangent at $P$ is the external bisector of $\angle SPS'$. $$\mathbf{\angle S P T = \angle S' P T}$$ Physical Consequence: Any light ray or acoustic wave emitted from one focus $S$ reflects off the elliptical boundary directly to the other focus $S'$! This is the physical mechanism of whispering galleries (such as St. Paul's Cathedral in London and the National Statuary Hall in Washington, D.C.).

2. Product of Perpendiculars from Foci onto Any Tangent

Let $p_1$ and $p_2$ be the lengths of perpendiculars dropped from the two foci $S(ae, 0)$ and $S'(-ae, 0)$ onto any tangent line $y - mx - \sqrt{a^2 m^2 + b^2} = 0$: $$p_1 = \frac{|-mae - \sqrt{a^2 m^2 + b^2}|}{\sqrt{1 + m^2}}, \qquad p_2 = \frac{|mae - \sqrt{a^2 m^2 + b^2}|}{\sqrt{1 + m^2}}$$ Multiplying the two perpendiculars: $$p_1 p_2 = \frac{|(a^2 m^2 + b^2) - m^2 a^2 e^2|}{1 + m^2} = \frac{|a^2 m^2(1 - e^2) + b^2|}{1 + m^2} = \frac{|a^2 m^2 (b^2/a^2) + b^2|}{1 + m^2} = \frac{b^2(m^2 + 1)}{1 + m^2} = \mathbf{b^2}$$ Theorem: The product of the perpendiculars from the foci onto any tangent to an ellipse is constant and equal to the square of the semi-minor axis $b^2$!