Conic Sections - The Algebraic Geometry of Slicing a Cone

The elegant curves known as conic sections—the parabola, ellipse, and hyperbola—have been a subject of mathematical fascination since ancient Greece. Originally studied as the shapes formed by slicing a double cone with a plane, they were later found to be central to describing physical phenomena, most famously Johannes Kepler's laws of planetary motion. Coordinate geometry provides a powerful framework to analyze these curves, translating their geometric properties into the precise language of algebra. This exploration delves into the unified definition of conics and examines the unique characteristics and equations of each type.

---

The Unifying Definition: Locus, Focus, Directrix, and Eccentricity

A conic section can be defined geometrically as a locus of points in a plane.

A conic section is the locus of a point $P$ that moves such that the ratio of its distance from a fixed point (the focus, $S$) to its perpendicular distance from a fixed straight line (the directrix) is a constant.

This constant ratio is called the eccentricity, denoted by $e$. $\dfrac{PS}{PM} = e$ where $PS$ is the distance from the point $P$ to the focus $S$, and $PM$ is the perpendicular distance from $P$ to the directrix line.

General Equation of a Conic Section

Let the focus be the point $S(h,k)$ and the directrix be the line $Ax+By+C=0$. Let any point on the conic be $P(x,y)$. From the definition, $PS^2 = e^2 PM^2$. $(x-h)^2 + (y-k)^2 = e^2 \left( \dfrac{|Ax+By+C|}{\sqrt{A^2+B^2}} \right)^2$ $(x-h)^2 + (y-k)^2 = \dfrac{e^2(Ax+By+C)^2}{A^2+B^2}$

Expanding this equation results in a general second-degree equation of the form $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$. where:

• $a = A^2(1-e^2) + B^2$
• $b = B^2(1-e^2) + A^2$
• $h = -ABe^2$
• $g = -ACe^2 - h(A^2 + B^2)$
• $f = -BCe^2 - k(A^2 + B^2)$
• $c = (A^2 + B^2)(h^2 + k^2) - C^2e^2$

---

Recognizing the Family: Types of Conic Sections and Their Recognition

The value of the eccentricity $e$ determines the type of the conic section:

• If $e=1$, the conic is a Parabola.
• If $0 < e < 1$, the conic is an Ellipse.
• If $e > 1$, the conic is a Hyperbola.
• If $e=0$, the conic is a Circle (which can be considered a special case of an ellipse where both foci coincide at the center).

These can also be recognized from the general equation of the second degree, $ax^2+2hxy+by^2+2gx+2fy+c=0$. The nature of the conic is determined by the values of $\Delta = abc+2fgh-af^2-bg^2-ch^2$ and the term $h^2-ab$.

• Parabola: if $\Delta \neq 0$ and $h^2-ab = 0$.
• Ellipse: if $\Delta \neq 0$ and $h^2-ab < 0$.
• Hyperbola: if $\Delta \neq 0$ and $h^2-ab > 0$.
• Circle: if $\Delta \neq 0$, $a=b$, and $h=0$.
• Pair of Straight Lines: if $\Delta = 0$.

---

The Parabola: The Perfect Arc ($e=1$)

Definition and Standard Equation

A parabola is the locus of a point equidistant from a fixed point (the focus) and a fixed line (the directrix). Let's derive the standard equation. Let the focus be $S(a,0)$ and the directrix be the line $x = -a$ (so $x+a=0$). Let $P(x,y)$ be any point on the parabola. By definition, $PS = PM$, where $PM$ is the perpendicular distance from $P$ to the line $x+a=0$. $PS^2 = PM^2$ $(x-a)^2 + (y-0)^2 = \left( \dfrac{|x+a|}{\sqrt{1^2+0^2}} \right)^2$ $x^2 - 2ax + a^2 + y^2 = (x+a)^2 = x^2 + 2ax + a^2$ $y^2 = 4ax$ This is the standard equation of a parabola opening to the right.

Terminology and Properties for $y^2 = 4ax$

• Axis of Symmetry: The line passing through the focus and perpendicular to the directrix. For $y^2=4ax$, this is the x-axis ($y=0$).
• Vertex: The point where the parabola intersects its axis. For $y^2=4ax$, this is the origin $(0,0)$.
• Focus: $S(a,0)$.
• Directrix: The line $x=-a$.
• Focal Chord: Any chord of the parabola that passes through the focus.
• Double Ordinate: A chord perpendicular to the axis of symmetry.
• Latus Rectum: The focal chord that is perpendicular to the axis of symmetry.
  • Length of Latus Rectum: To find the endpoints, set $x=a$ in $y^2=4ax$. $y^2 = 4a(a) = 4a^2 \implies y = \pm 2a$. The endpoints are $(a, 2a)$ and $(a, -2a)$. The length is the distance between these points, which is $4a$.
• Parametric Form: A point on the parabola can be represented as $(at^2, 2at)$, where $t$ is a parameter. The parameter $t$ represents the slope of the tangent at that point.

Other Standard Forms

• $y^2 = -4ax$: Opens left, Focus $(-a,0)$, Directrix $x=a$.
• $x^2 = 4ay$: Opens up, Focus $(0,a)$, Directrix $y=-a$.
• $x^2 = -4ay$: Opens down, Focus $(0,-a)$, Directrix $y=a$.
• General Form (Shifted Vertex): For a parabola with vertex at $(h,k)$ and axis parallel to the x-axis, the equation is $(y-k)^2 = 4a(x-h)$.
• Parametric Form: For a parabola with vertex at $(h,k)$, the parametric equations are: $x = h + at^2, \quad y = k + 2at$

Intersection with Lines: Tangents and Normals

1. Tangent to a Parabola:

   • Point Form: The equation of the tangent to the parabola $y^2=4ax$ at a point $(x_1, y_1)$ on it is: $T \equiv 0 \implies yy_1 - 2a(x+x_1) = 0$ Derivation: Differentiate $y^2=4ax$ implicitly with respect to $x$: $2y \dfrac{dy}{dx} = 4a \implies \dfrac{dy}{dx} = \dfrac{2a}{y}$. At $(x_1, y_1)$, the slope of the tangent is $m = 2a/y_1$. Using point-slope form $y-y_1 = m(x-x_1)$ gives $y-y_1 = \dfrac{2a}{y_1}(x-x_1) \implies yy_1 - y_1^2 = 2ax - 2ax_1$. Since $y_1^2 = 4ax_1$, we have $yy_1 - 4ax_1 = 2ax - 2ax_1$, which simplifies to the result.
   • Slope Form:

     Equation	Point of Contact $P(m)$	Tangent at $P(m)$	Condition of tangency
     $y^2 = 4ax$	$(\dfrac{a}{m^2}, \dfrac{2a}{m})$	$y = mx + \dfrac{a}{m}$	$c = \dfrac{a}{m}$
     $y^2 = -4ax$	$(-\dfrac{a}{m^2}, -\dfrac{2a}{m})$	$y = mx - \dfrac{a}{m}$	$c = -\dfrac{a}{m}$
     $x^2 = 4ay$	$(2am, am^2)$	$y = mx - am^2$	$c = -am^2$
     $x^2 = -4ay$	$(-2am, -am^2)$	$y = mx + am^2$	$c = am^2$

   • Pair of Tangents from a Point: For a point $(x_1, y_1)$ outside the parabola, the equation of the pair of tangents is: $SS_1 = T^2$
   • The chord of contact from a point $(x_1, y_1)$ outside the parabola is given by: $yy_1 = 2a(x+x_1)$
   • Properties of Tangents:
     • Point of intersection of tangents at any two points $P (at_1^2, 2at_1)$ and $Q (at_2^2, 2at_2)$ is given by $(at_1t_2, a(t_1+t_2))$, i.e., x-coordinate is the geometric mean of the x-coordinates of $P$ and $Q$, and y-coordinate is the arithmetic mean of the y-coordinates.
     • Locus of foot of perpendiculars from the focus upon any tangent is tangent to the parabola at the vertex.
     • Length of tangent between the point of contact and the point where it meeets the directrix subtends a right angle at the focus.
     • Tangents at extremities of a focal chord are perpendicular and intersect at the directrix.

2. Normal to a Parabola: The normal is the line perpendicular to the tangent at the point of contact.

   • Point Form at $(x_1, y_1)$: The slope of the normal is $-1/m_{tangent} = -y_1/(2a)$. The equation is: $y-y_1 = -\dfrac{y_1}{2a}(x-x_1)$
   • Slope Form:

     Equation	Point of Contact $P(m)$	Normal at $P(m)$	Condition of normalcy
     $y^2 = 4ax$	$(am^2, -2am)$	$y = mx - 2am - am^3$	$c = - 2am - am^3$
     $y^2 = -4ax$	$(-am^2, 2am)$	$y = mx + 2am + am^3$	$c = 2am + am^3$
     $x^2 = 4ay$	$(-\dfrac{2a}{m}, \dfrac{a}{m^2})$	$y = mx + 2a + \dfrac{a}{m^2}$	$c = 2a + \dfrac{a}{m^2}$
     $x^2 = -4ay$	$(\dfrac{2a}{m}, -\dfrac{a}{m^2})$	$y = mx - 2a - \dfrac{a}{m^2}$	$c = - 2a - \dfrac{a}{m^2}$

   • Properties of Normals:
     • Normal other than axis of parabola never passes through the focus.
     • Point of intersection of normals at any two points $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ is given by $(2a+a(t_1^2+t_2^2+t_1t_2), -at_1t_2(t_1+t_2))$.
     • Normal at the poiny $P(t_1)$ meets the curve again at point $Q(t_2)$, such that $t_1t_2 + t_1^2 + 2 = 0$.

3. Parametric Form of Tangent and Normal:

   Equation	$P(t)$	Tangent at $P(t)$	Normal at $P(t)$
   $y^2 = 4ax$	$(at^2, 2at)$	$ty = x + at^2$	$y + tx = 2at + at^3$
   $y^2 = -4ax$	$(-at^2, 2at)$	$ty = -x + at^2$	$y - tx = 2at + at^3$
   $x^2 = 4ay$	$(2at, at^2)$	$tx = y + at^2$	$x + ty = 2at + at^3$
   $x^2 = -4ay$	$(2at, -at^2)$	$tx = -y + at^2$	$x - ty = 2at + at^3$

4. Focal Chord: A chord passing through the focus. The equation of the focal chord at $(x_1, y_1)$ is: $y - y_1 = \dfrac{y_1}{x_1 - a}(x - x_1)$

   • Properties of Focal Chords:
     • If the chord joining $P \equiv (at_1^2, 2at_1)$ and $Q \equiv (at_2^2, 2at_2)$ is the focal chord then $t_1t_2 = -1$.
     • If point $P \equiv (at^2, 2at)$, then length of focal chord $PQ$ is $a\left(t+\dfrac{1}{t}\right)^2$
     • The length of the focal chord which makes an angle $\theta$ with positive direction of the x-axis is $4a\cosec^2\theta$.
     • Semi-latus rectum is harmonic mean of $SP$ and $SQ$, where $S$ is the focus and $P$ and $Q$ are the endpoints of the focal chord.
     • Circle described on the focal chord as diameter touches directix & the circle described on the focal length as diameter touches the tangent at vertex.

Properties and Reflection

• Co-normal Points: From any given point $(h,k)$, three normals can be drawn to a parabola (in general). The points on the parabola where these normals are drawn are called co-normal points.
  • The algebraic sum of ordinates of the feet of three normals drawn to a parabola from a given point is 0.
  • Centroid of the triangle formed by the feet of the three normals lies on the axis ofthe parabola.
  • If three normals drawn to any parabola $y^2 = 4ax$ from a given point $(h, k)$ be real, then $h \gt 2a$.
  • Circle through the co-normal points is given by the equation: $x^2 + y^2 - (2a + h)x - \dfrac{k}{2}y = 0$
• Reflection Property: A ray of light traveling parallel to the axis of symmetry of a parabola is reflected through the focus. This property is fundamental to the design of parabolic reflectors used in satellite dishes, telescopes, and headlights.

---

The Ellipse: The Cosmic Orbit ($0 < e < 1$)

Definition and Standard Equation

An ellipse can be defined in two common ways:

1. Focus-Directrix: The locus of a point $P$ such that its distance from a fixed point (focus $S$) is a constant multiple $e$ (where $0<e<1$) of its distance from a fixed line (directrix). $PS = ePM$.
2. Two Foci: The locus of a point $P$ such that the sum of its distances from two fixed points (the foci, $S_1$ and $S_2$) is a constant ($2a$). $PS_1 + PS_2 = 2a$.

Let's derive the standard equation from the two-foci definition. Let the foci be $S_1(-c,0)$ and $S_2(c,0)$, and the constant sum be $2a$. For any point $P(x,y)$ on the ellipse: $\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a$. Rearranging and squaring twice leads to: $\dfrac{x^2}{a^2} + \dfrac{y^2}{a^2-c^2} = 1$ We define $b^2 = a^2 - c^2$. Since for an ellipse $a>c$, $b^2$ is positive. $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ This is the standard equation for an ellipse centered at the origin with its major axis along the x-axis.

Terminology and Properties for $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a>b$)

• Center: The midpoint of the foci, $(0,0)$.
• Axes of Symmetry: The major axis (length $2a$, along the x-axis) and the minor axis (length $2b$, along the y-axis).
• Vertices: Endpoints of the major axis, $(\pm a, 0)$.
• Foci: The two fixed points, $S_1(-c,0)$ and $S_2(c,0)$.
• Eccentricity ($e$): The ratio $e = c/a$. Since $c = \sqrt{a^2-b^2}$, we have $e = \dfrac{\sqrt{a^2-b^2}}{a}$. This leads to the important relation: $b^2 = a^2(1-e^2)$
• Directrices: The two fixed lines corresponding to the foci. Their equations are $x = \pm a/e$.
• Latus Rectum: A focal chord perpendicular to the major axis.
  • Length: Set $x=ae$ in the ellipse equation to find the y-coordinates: $\dfrac{(ae)^2}{a^2} + \dfrac{y^2}{b^2} = 1 \implies e^2 + \dfrac{y^2}{b^2} = 1 \implies y^2 = b^2(1-e^2) = b^2(b^2/a^2) = b^4/a^2$. So $y = \pm b^2/a$. The total length is $2b^2/a$.
• Properties of the Ellipse:
  • The area of the ellipse is $\pi ab$.
  • Ratio of area of any triangle $PQR$ inscribed in the ellipse and that of triangle formed by corresponding points on the auxiliary circle is $b/a$.
  • Semi latus rectum is harmonic mean of segments of focal chord or $\dfrac{1}{SP} + \dfrac{1}{SQ} = \dfrac{2a}{b^2} (a \gt b)$ (where PQ is focal chord through focus S).
  • Circle described on focal length as diameter always touches auxiliary circle.

Auxiliary Circle and Eccentric Angle

• Auxiliary Circle: The circle with the major axis of the ellipse as its diameter. Its equation is $x^2+y^2=a^2$.
• Eccentric Angle ($\phi$): Any point on the ellipse can be represented parametrically as $(a\cos\phi, b\sin\phi)$. The angle $\phi$ is called the eccentric angle of the point. It is the angle that the line from the center to the corresponding point on the auxiliary circle makes with the positive x-axis.

Intersection with Lines: Tangents and Normals

1. Tangent to an Ellipse:

   • Point Form at $(x_1, y_1)$: $\dfrac{xx_1}{a^2} + \dfrac{yy_1}{b^2} = 1$
   • Slope Form: $y = mx \pm \sqrt{a^2m^2 + b^2}$. The condition for tangency for line $y=mx+c$ is $c^2 = a^2m^2+b^2$.
   • Parametric Form at $(a\cos\phi, b\sin\phi)$: $\dfrac{x\cos\phi}{a} + \dfrac{y\sin\phi}{b} = 1$
   • Pair of Tangents from a Point: For a point $(x_1, y_1)$ outside the ellipse, the equation of the pair of tangents is: $SS_1 = T^2$
   • Point of Contact where line $y = mx + c$ touches the Ellipse is given by: $(\pm \dfrac{a^2m}{\sqrt{a^2m^2 + b^2}}, \mp \dfrac{b^2}{\sqrt{a^2m^2 + b^2}})$
   • Properties of Tangents:
     • For ellipse $x^2/a^2 + y^2/b^2 = 1$, the equation of tangent at anu point $\theta$, i.e., at $(a\cos\theta, b\sin\theta)$ is: $\dfrac{x\cos\theta}{a} + \dfrac{y\sin\theta}{b} = 1$
     • Product of perpendiculars from foci upon any tangent of the ellipse is $b^2$.
     • Tangents at the extremities of latus rectum pass through the corresponding foot of directrix on major axis.
     • Length of tangent between the point of contact and the point where it meets the directrix subtends right angle at the corresponding focus.
     • Point of intersection of tangents at any two points $P(\alpha)$ and $Q(\beta)$ is given by: $\left( a\dfrac{\cos\dfrac{\alpha+\beta}{2}}{\cos\dfrac{\alpha-\beta}{2}}, b\dfrac{\sin\dfrac{\alpha+\beta}{2}}{\sin\dfrac{\alpha-\beta}{2}} \right)$

2. Normal to an Ellipse:

   • Point Form at $(x_1, y_1)$: $\dfrac{a^2x}{x_1} - \dfrac{b^2y}{y_1} = a^2 - b^2$

Properties and Reflection

• Co-normal Points: In general, four normals can be drawn from a point to an ellipse. The condition for their eccentric angles is $\alpha + \beta + \gamma + \delta = (2n + 1)\pi$.
• Concyclic Points: The points of intersection of an ellipse and a circle are, in general, four points. These are concyclic by definition. $\alpha + \beta + \gamma + \delta = 2n\pi, n \in \mathbb{Z}$.
• Reflection Property: A ray originating from one focus of an ellipse will reflect off the ellipse and pass through the other focus. This is used in "whispering galleries."

---

The Hyperbola: The Scattering Path ($e>1$)

Definition and Standard Equation

A hyperbola can be defined in two common ways:

1. Focus-Directrix: The locus of a point $P$ such that its distance from a focus $S$ is a constant multiple $e$ (where $e>1$) of its distance from a directrix line. $PS = ePM$.
2. Two Foci: The locus of a point $P$ such that the absolute difference of its distances from two fixed points (the foci, $S_1$ and $S_2$) is a constant ($2a$). $|PS_1 - PS_2| = 2a$.

Deriving from the two-foci definition, with foci at $(\pm c, 0)$ and constant difference $2a$, leads to the standard equation: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ where we define $b^2 = c^2 - a^2$. Since $e=c/a > 1$, $c>a$, so $b^2$ is positive.

Terminology and Properties for $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$

• Center: The midpoint of the foci, $(0,0)$.
• Axes: The transverse axis is the line segment connecting the vertices (length $2a$). The conjugate axis is a line segment of length $2b$ perpendicular to the transverse axis at the center.
• Vertices: Endpoints of the transverse axis, $(\pm a, 0)$.
• Foci: $(\pm c, 0)$ or $(\pm ae, 0)$.
• Eccentricity ($e$): $e = c/a > 1$. The key relation is: $b^2 = a^2(e^2 - 1)$
• Directrices: The lines $x = \pm a/e$.
• Latus Rectum: Length is $2b^2/a$.
• Asymptotes: Two straight lines that the hyperbola approaches as it tends to infinity. The equations of the asymptotes are: $y = \pm \dfrac{b}{a}x \quad \text{or} \quad \dfrac{x}{a} \pm \dfrac{y}{b} = 0$

Special Types of Hyperbolas

• Rectangular Hyperbola: A hyperbola whose asymptotes are perpendicular. This occurs when $a=b$. Its equation is $x^2-y^2=a^2$, and its eccentricity is $e = \sqrt{2}$. When rotated by $45^\circ$, its equation can be written as $xy=k$.
• Conjugate Hyperbola: The hyperbola with equation $\dfrac{y^2}{b^2} - \dfrac{x^2}{a^2} = 1$. It has the same center and asymptotes as the original hyperbola, but its transverse axis is along the y-axis.

Auxiliary Circle and Eccentric Angle

• Auxiliary Circle: The circle with the transverse axis as its diameter, $x^2+y^2=a^2$.
• Eccentric Angle ($\phi$): The parametric representation of a point on the hyperbola is $(a\sec\phi, b\tan\phi)$.

Intersection with Lines: Tangents and Normals

1. Tangent to a Hyperbola:

   • Point Form at $(x_1, y_1)$: $\dfrac{xx_1}{a^2} - \dfrac{yy_1}{b^2} = 1$
   • Slope Form: $y = mx \pm \sqrt{a^2m^2 - b^2}$. Condition for tangency for $y=mx+c$ is $c^2=a^2m^2-b^2$.
   • Parametric Form at $(a\sec\phi, b\tan\phi)$: $\dfrac{x\sec\phi}{a} - \dfrac{y\tan\phi}{b} = 1$

2. Normal to a Hyperbola:

   • Point Form at $(x_1, y_1)$: $\dfrac{a^2x}{x_1} + \dfrac{b^2y}{y_1} = a^2 + b^2 = a^2e^2$

Properties

• Co-normal and Concyclic Points: Concepts are analogous to the ellipse.
• Reflection Property: A ray directed towards one focus of a hyperbola will reflect off the hyperbola as if it originated from the other focus.

---

Key Takeaways: The Unified Geometry of Conics

Conic sections represent a beautiful unification of geometric and algebraic concepts, describing a family of related curves with diverse yet interconnected properties.

• Unified Definition: All non-degenerate conic sections (parabola, ellipse, hyperbola) can be defined as the locus of a point whose distance from a fixed point (focus) is in a constant ratio (eccentricity, $e$) to its distance from a fixed line (directrix).
• Eccentricity as a Classifier: The value of eccentricity ($e=1$ for parabola, $0<e<1$ for ellipse, $e>1$ for hyperbola) provides a single parameter that defines the fundamental nature and shape of the conic.
• Standard Equations: Each conic has a standard algebraic form when its axes are aligned with the coordinate axes. These equations and their associated parameters (vertex, foci, axes, latus rectum) provide a complete description of the curve.
• Tangent and Normal Properties: The equations of tangents and normals to conics can be systematically determined in various forms (point, slope, parametric), revealing consistent structural patterns across the different types.
• Physical Significance: Beyond their geometric elegance, conic sections are fundamental to the physical world, describing the paths of objects under inverse-square law forces, such as planetary orbits (ellipses), paths of thrown objects (parabolas), and scattering trajectories of comets (hyperbolas).

The study of conic sections through coordinate geometry is a classic example of the power of algebra to illuminate, quantify, and solve complex geometric problems, providing tools essential to physics, astronomy, and engineering.