Conic Sections

Conic sections represent the curves obtained by slicing a three-dimensional double-napped right circular cone by an intersecting flat plane at various angles of inclination.

Let a plane intersect a double right circular cone of semi-vertical angle $\alpha$ at an angle $\beta$ with the vertical cone axis:

• Circle: When $\beta = 90^\circ$ (plane is perpendicular to the cone axis), the section is a Circle.
• Ellipse: When $\alpha < \beta < 90^\circ$ (plane cuts through a single nappe entirely), the section is an Ellipse.
• Parabola: When $\beta = \alpha$ (plane is parallel to a generator/slant height of the cone), the section is an unbounded Parabola.
• Hyperbola: When $0 \le \beta < \alpha$ (plane is parallel to the cone axis or cuts both nappes), the section forms a two-branched Hyperbola.
• Degenerate Conics: If the intersecting plane passes through the apex (vertex) of the cone:
  • $\beta > \alpha$: The section degenerates into a single Point.
  • $\beta = \alpha$: The section degenerates into a single Straight Line (generator).
  • $\beta < \alpha$: The section degenerates into a Pair of Intersecting Straight Lines.

Let focus be $S(x_0, y_0)$, directrix be $ax + by + c = 0$, and eccentricity be $e$. For any point $P(x, y)$ on the conic:

A circle is the locus of all points in a plane equidistant from a fixed center. Analytically, it is a conic with zero eccentricity.

• Central Form: A circle with center at the origin $O(0, 0)$ and radius $r$: $$\mathbf{x^2 + y^2 = r^2}$$
• Standard (Center-Radius) Form: A circle with center $C(h, k)$ and radius $r$: $$\mathbf{(x - h)^2 + (y - k)^2 = r^2}$$

Expanding $(x - h)^2 + (y - k)^2 = r^2$ gives $x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0$. Setting $g = -h$, $f = -k$, and $c = h^2 + k^2 - r^2$ leads to:

Nature of the Circle depending on $g^2 + f^2 - c$:

• If $g^2 + f^2 - c > 0$: The circle is a Real Circle with non-zero radius.
• If $g^2 + f^2 - c = 0$: The circle is a Point Circle (radius is zero, representing only the single point $(-g, -f)$).
• If $g^2 + f^2 - c < 0$: The circle is an Imaginary (Virtual) Circle with no real Cartesian points.

The general quadratic equation $Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0$ represents a circle if and only if:

1. $\mathbf{A = B \neq 0}$ (Coefficients of $x^2$ and $y^2$ are equal).
2. $\mathbf{H = 0}$ (Coefficient of the cross product term $xy$ is zero).
3. $\mathbf{G^2 + F^2 - AC \ge 0}$ (Radius is real).

If $A(x_1, y_1)$ and $B(x_2, y_2)$ are the endpoints of a diameter of a circle, then for any point $P(x, y)$ on the circumference, $\angle APB = 90^\circ$ (angle in a semicircle). Hence the slopes satisfy $m_{AP} \cdot m_{BP} = -1$:

For a circle $(x - h)^2 + (y - k)^2 = r^2$, every point on the circumference can be parameterized by the angle $\theta \in [0, 2\pi)$:

For the central circle $x^2 + y^2 = r^2$, the parametric equations simplify to $\mathbf{x = r\cos\theta, \; y = r\sin\theta}$.

A parabola is the conic section with eccentricity $e = 1$. It represents the locus of points equidistant from a fixed focus and a fixed directrix line.

• Focus ($S$): The fixed point.
• Directrix ($L$): The fixed line perpendicular to the axis of symmetry.
• Axis: The line passing through the focus and perpendicular to the directrix.
• Vertex ($V$): The midpoint of the segment joining the focus and the directrix along the axis.
• Focal Chord: Any line segment passing through the focus and terminating on the parabola.
• Latus Rectum ($LL'$): The focal chord perpendicular to the axis of symmetry.
• Focal Distance ($SP$): The distance of any point $P(x, y)$ on the parabola from the focus.

When the vertex is translated from origin to $V(h, k)$ while maintaining axes parallel to coordinate axes:

• Axis parallel to $x$-axis: $\mathbf{(y - k)^2 = 4a(x - h)}$ (Focus: $(h + a, k)$, Directrix: $x = h - a$).
• Axis parallel to $y$-axis: $\mathbf{(x - h)^2 = 4a(y - k)}$ (Focus: $(h, k + a)$, Directrix: $y = k - a$).

An ellipse is the conic section with eccentricity $0 < e < 1$. It is defined either through the focus-directrix property or as the locus of points whose sum of distances from two fixed foci is constant.

The circle described on the major axis of an ellipse as diameter is called its auxiliary circle, having equation $x^2 + y^2 = a^2$.

The parametric equations of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are:

A hyperbola is the conic section with eccentricity $e > 1$. It features two disconnected open branches and is defined by constant difference between focal distances.

An asymptote is a straight line that touches the hyperbola at infinity without ever intersecting it at any finite point. For the standard hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$:

A hyperbola in which the length of the transverse axis equals the length of the conjugate axis ($a = b$) is called a Rectangular Hyperbola:

• Standard Equation: $\mathbf{x^2 - y^2 = a^2}$
• Eccentricity: $e = \sqrt{1 + \frac{a^2}{a^2}} = \sqrt{1 + 1} = \mathbf{\sqrt{2}}$ (constant for ALL rectangular hyperbolas!).
• Asymptotes: $y = \pm x \implies x - y = 0$ and $x + y = 0$, which are mutually perpendicular (hence "rectangular").
• Rotated Form (referred to asymptotes as coordinate axes): $\mathbf{xy = c^2}$ where $c^2 = a^2/2$.

Using the identity $\sec^2 \theta - \tan^2 \theta = 1$, the parametric equations of $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are:

Any second-degree polynomial equation in two variables represents a conic section. Understanding the algebraic discriminant unlocks immediate geometric identification.

Consider the general quadratic equation in $x$ and $y$:

Define the algebraic determinant discriminant $\Delta$:

• Parabolic Reflection: Any ray originating from the focus reflects off the interior surface parallel to the axis of symmetry. Conversely, any incoming ray parallel to the axis reflects directly through the focus. Applications: Car headlights, parabolic satellite dishes, astronomical reflecting telescopes.
• Elliptic Reflection: Any ray emitted from one focus $S$ reflects off the inner elliptical boundary and passes directly through the second focus $S'$. Applications: Whispering galleries (St. Paul Cathedral), medical lithotripter machines for shattering kidney stones.
• Hyperbolic Reflection: A ray directed toward one external focus $S'$ reflects off the exterior hyperbolic branch along a straight trajectory whose backward extension passes through the internal focus $S$. Applications: Cassegrain telescope secondary mirrors, radio signal hyperbolic multilateration (LORAN).