Conic Sections

Conic Sections are curves obtained by intersecting a cone with a plane in different orientations. These curves are fundamental in geometry and have significant applications in physics, astronomy, architecture, and many other fields. The four classical conic sections are:

1. Circunference
2. Ellipse
3. Parabola
4. Hyperbola

Definition and Classification

Imagine a double cone, i.e., two cones joined at their vertices, one pointing upwards and the other downwards. By intersecting this cone with a plane, depending on the angle of inclination of the plane relative to the axis of the cone, we obtain different curves:

1. Circunference: If the plane is perpendicular to the cone’s axis (i.e., parallel to the cone's base), the intersection produces a circunference.
2. Ellipse: When the plane cuts through the cone at an angle less than that of the cone's generating lines (but does not pass through the vertex), it forms an ellipse. A circunference is a special case of an ellipse where both axes are equal.
3. Parabola: If the plane is parallel to a generating line of the cone (i.e., it touches the cone in just one line), a parabola is formed. This is the only conic section that extends infinitely in both directions.
4. Hyperbola: If the plane cuts through both halves of the cone (the two opposite cones), forming an angle less than that of the generating lines, it results in a hyperbola, which has two distinct branches.

Adjust the characteristics of the conics sections by moving the sliders.

Circunference

All points are at a constant distance (radius) from the center.

Equations

The equation for the circunference is

$$\left(x -h\right)^{2} + \left(y - k\right)^{2} = r^{2}$$

To obtain the general form it is necessary to make the following substitutions:

$$D = -2h,\ E = -2k,\\ F = h^{2} + k^{2} - r^{2}$$

Then,

$$\begin{split}x^{2} + y^{2} + Dx + Ey + F &= 0 \\ \Rightarrow x^{2} + y^{2} - 2hx - 2ky &= -F \\ \Rightarrow x^{2} - 2hx + h^{2} + y^{2} - 2ky + k^{2} &= \\ - F + h^{2} + k^{2} &= \\ \Rightarrow \left(x - h\right)^{2} + \left(y - k\right)^{2} &= r^{2} \end{split}$$

The equation of a circunference from a segment that forms the diameter is:

$$\left(x - x_{1}\right)\left(x - x_{2}\right) + \left(y - y_{1}\right)\left(y - y_{2}\right) = 0$$

Ellipse

It has two foci. The sum of the distances from any point on the ellipse to the two foci is constant.

Equations

$$\frac{\left( x - h \right)^{2}}{a^{2}} + \frac{\left( y - k \right)^{2}}{b^{2}} = 1$$

$$c = \sqrt{b^{2} - a^{2}}$$

Parabola

It has a focus and a directrix. Any point on the parabola is equidistant from the focus and the directrix.

Equations

Vertical

$$\left( y - k \right)^{2} = 4p \left( x - h \right)$$

Horizontal

$$\left( x - h \right)^{2} = 4p \left( y - k \right)$$

Demonstration

We take any point on the parabola with coordinates \((x,y)\) and calculate its distance to the focus and the directrix.

Distance from the point to the focus

$$d_{1} = \sqrt{\left( x - h \right)^{2} + \left( y - \left( k + p \right) \right)^{2}}$$

Distance from the point to the directrix

$$d_{2} = \sqrt{\left( x - x \right)^{2} + \left( y - \left( k - p \right) \right)^{2}}$$

To simplify, we have \(d_{2} = \left( y - \left( k - p \right) \right)\), and since \(d_{1}=d_{2}\) by the definition of the parabola.

$$\sqrt{\left( x - h \right)^{2} + \left( y - \left( k + p \right) \right)^{2}}\\ = \left( y - \left( k - p \right) \right)$$

We square both sides.

$$\begin{split}\left( x - h \right)^{2} &+ \left( y - \left( k + p \right) \right)^{2}\\ &= \left( y - \left( k - p \right) \right)^{2}\end{split}$$

Since there are only repeated terms of \(y\), we expand the binomials.

$$\begin{split}\left( x - h \right)^{2} &+ y^{2} - 2y \left( k + p \right) + \left( k + p \right)^{2}\\ &= y^{2} - 2y \left( k - p \right) + \left( k - p \right)^{2}\end{split}$$

We cancel the repeated terms on both sides and group the terms containing \(y\) on the right-hand side.

$$\begin{split}\left( x - h \right)^{2} &= 2y \left( k + p \right) \\ &- 2y \left( k - p \right) \\ &+ \left( k - p \right)^{2} \\ &- \left( k + p \right)^{2}\end{split}$$

$$\begin{split}\left( x - h \right)^{2} &= 2y \left( k + p - k + p \right) \\ &+ k^{2} - 2kp + p^{2} \\ &- \left( k^{2} + 2kp + p^{2} \right)\end{split}$$

$$\left( x - h \right)^{2} = 2y\left( 2p \right) - 4kp$$

$$\therefore \left( x - h \right)^{2} = 4p\left( y - k \right)$$

Hyperbola

It has two branches, each approaching two asymptotes.

The difference in distances from any point on the hyperbola to the two foci is constant.

Equations

Vertical

$$\frac{\left( x - h \right)^{2}}{a^{2}} - \frac{\left( y - k \right)^{2}}{b^{2}} = - 1$$

Horizontal

$$\frac{\left( x - h \right)^{2}}{a^{2}} - \frac{\left( y - k \right)^{2}}{b^{2}} = 1$$

General Form

Equation

Conic sections can be described using quadratic equations in a Cartesian coordinate system. The general form is:

$$Ax^{2} + Bxy + Cy^{2} + Dx + Ex + F = 0$$

Depending on the values of \(A\), \(B\), and \(C\), the equation can represent different conic sections:

1. circunference: If \(A = C\) and \(B = 0\).
2. Ellipse: If \(A \neq C\), \(B = 0\), and both coefficients have the same sign \( \left(A \cdot C > 0\right) \).
3. Parabola: If \(AC = 0\) and \(B = 0\), or if the discriminant of the quadratic form is zero.
4. Hyperbola: If \(A\) and \(C\) have opposite signs \( \left( A \cdot C < 0 \right) \).

Applications of Conic Sections

1. Circunference:Circles: Used in the design of wheels, gears, and circular orbits.
2. Ellipses: The orbits of planets and satellites are elliptical, as described by Kepler’s laws.
3. Parabolas: Trajectories of projectiles and the design of parabolic reflectors (like satellite dishes and headlights).
4. Hyperbolas: Used in the design of telescopes, the trajectories of comets, and GPS navigation systems.

See also

Line equations

Mirrors

Quadric Surfaces