Comprehensive Guide to Conic Sections

Fundamental Formulas and Circles

In the study of conic sections, several fundamental algebraic formulas serve as the basis for calculations. The Midpoint Formula, used to find the center point between two coordinates $(x_1, y_1)$ and $(x_2, y_2)$, is defined as $(\frac{x_1 + x_2}{2}, \frac{y_1 + y^2}{2})$. The Distance Formula, which calculates the length between two points in a Cartesian plane, is expressed as $d = \sqrt{(x_2 - x_1)^2 + (\frac{1}{2} - y_1)^2}$. These formulas are often applied to find centers, radii, or axis lengths for various conics.

A circle is defined by the set of all points equidistant from a central point $(h, k)$. The standard equation for a circle is given as $(x - h)^2 + (y - k)^2 = p^2$, where the radius is represented by $r$. In an alternative fractional form, the equation can be written as $\frac{(x - h)^2}{r^2} + \frac{(y - k)^2}{r^2} = 1$. The diameter of the circle is twice the radius, or $d = 2r$. The domain of a circle, representing the range of horizontal values, is defined by the interval $[h - r, h + r]$, while the range, representing the vertical values, is defined by the interval $[k - r, k + r]$.

Properties of Ellipses

Ellipses are categorized by their orientation, which can be either horizontal or vertical. For all ellipses, the center is located at $(h, k)$. The relationship between the lengths defining the ellipse's shape and the distance to its foci is governed by the equation $b^2 + c^2 = a^2$. In this context, $a$ represents the distance from the center to the vertices along the major axis, $b$ represents the distance to the co-vertices along the minor axis, and $c$ represents the distance to the foci.

A horizontal ellipse is defined by the equation $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$. The major axis is horizontal with a total length of $2a$ units, while the minor axis is vertical with a length of $2b$ units. The vertices of a horizontal ellipse are positioned at $(h \pm a, k)$, and the co-vertices are at $(h, k \pm b)$. Its foci are located at $(h \pm c, k)$. The domain is constrained to $[h - a, h + a]$ and the range is constrained to $[k - b, k + b]$.

A vertical ellipse is defined by the equation $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$. The major axis is vertical with a length of $2a$ units, and the minor axis is horizontal with a length of $2b$ units. The vertices are found at $(h, k \pm a)$ and the co-vertices are located at $(h \pm b, k)$. The foci are positioned at $(h, k \pm c)$. The domain of a vertical ellipse is $[h - b, h + b]$ and the range is $[k - a, k + a]$.

Properties of Hyperbolas

Hyperbolas consist of two separate curves called branches and can be oriented horizontally or vertically. The center for both types is $(h, k)$. The relationship between variables $a$, $b$, and $c$ for hyperbolas is defined by the formula $a^2 + b^2 = c^2$. Here, $a$ is the distance from the center to each vertex, and $c$ is the distance from the center to each focus.

A horizontal hyperbola follows the equation $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$. The transverse axis is horizontal with a length of $2a$ units, and the conjugate axis is vertical with a length of $2b$ units. The vertices are located at $(h \pm a, k)$ and the foci are at $(h \pm c, k)$. The asymptotes, which the curves approach but never touch, are defined by the equation $y - k = \pm \frac{b}{a}(x - h)$. The domain for this conic is $(-\infty, h - a] \cup [h + a, \infty)$, and the range is $(-\infty, \infty)$.

A vertical hyperbola follows the equation $\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1$. The transverse axis is vertical with a length of $2a$ units, and the conjugate axis is horizontal with a length of $2b$ units. The vertices are located at $(h, k \pm a)$ and the foci are at $(h, k \pm c)$. The asymptotes are given by the linear equation $y - k = \pm \frac{a}{b}(x - h)$. The domain for vertical hyperbolas is $(-\infty, \infty)$, and the range is $(-\infty, k - a] \cup [k + a, \infty)$.

Properties of Parabolas

Parabolas are defined by an equation in which only one variable is squared, and they can open horizontally (left or right) or vertically (up or down). The vertex, or the turning point of the parabola, is always at $(h, k)$. The distance from the vertex to the focus and from the vertex to the directrix is represented by the value $p$. The Latus Rectum (LR) is a chord through the focus perpendicular to the axis of symmetry, with a length defined as $|4p|$ units.

Parabolas that open left or right are defined by the equation $(y - k)^2 = 4p(x - h)$. The axis of symmetry is the horizontal line $y = k$. The focus is located at $(h + p, k)$, and the directrix is the vertical line $x = h - p$. The latus rectum is vertical. If the value of p > 0, the parabola opens to the right; if p < 0, it opens to the left. For parabolas opening right, the domain is $[h, \infty)$, and for those opening left, the domain is $(-\infty, h]$. The range is always $(-\infty, \infty)$.

Parabolas that open up or down follow the equation $(x - h)^2 = 4p(y - k)$. The axis of symmetry is the vertical line $x = h$. The focus is located at $(h, k + p)$, and the directrix is the horizontal line $y = k - p$. The latus rectum is horizontal. If p > 0, the parabola opens up; if p < 0, it opens down. In these cases, the domain is $(-\infty, \infty)$. The range is $[k, \infty)$ if p > 0 and $(-\infty, k]$ if p < 0.

Identification of Conics from General Form

Any conic section can be represented in the general form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. The values of the coefficients $A$, $B$, and $C$ are the primary indicators of the type of conic section described by the equation. For current problems, the value of $B$ is assumed to be $0$, meaning the classification rests solely on the values of $A$ and $C$.

If either $A = 0$ or $C = 0$, the conic is a parabola. This is because parabolas are characterized by having only one squared term. If $A = C$, the conic is a circle. If the coefficients $A$ and $C$ have the same sign but are not equal to each other (for example, both are positive or both are negative), the conic is an ellipse. Finally, if $A$ and $C$ have opposite signs (one is positive and the other is negative), the conic is identified as a hyperbola.