Applied Mathematics Essentials

Cheatsheet Content

Prime Numbers & Factorization Divisibility: $a|b$ if $b = ac$ for $c \in \mathbb{Z}$. Prime: A positive integer $p > 1$ with only positive divisors $1$ and $p$. (e.g., $2, 3, 5, 7, \dots$) Composite: A positive integer $n > 1$ that is not prime. (e.g., $4, 6, 8, \dots$) Fundamental Theorem of Arithmetic: Every $n > 1$ can be uniquely written as $n = p_1^{a_1} p_2^{a_2} \dots p_r^{a_r}$ where $p_i$ are distinct primes and $a_i$ are positive integers. Euclid's Theorem: There are infinitely many prime numbers. Sieve of Eratosthenes: Method to find primes up to $n$ by eliminating multiples of primes $\le \sqrt{n}$. Cryptography (RSA) GCD (Greatest Common Divisor): $d = \text{gcd}(a, b)$ is the largest positive integer that divides both $a$ and $b$. Relatively Prime/Coprime: $a, b$ are coprime if $\text{gcd}(a, b) = 1$. Euclid's GCD Algorithm: For $b = aq + r$, $\text{gcd}(a, b) = \text{gcd}(a, r)$. Repeat until $r=0$, the last non-zero $r$ is the GCD. Modular Arithmetic: $a \equiv b \pmod{m}$ if $m | (a-b)$. Means $a$ and $b$ have the same remainder when divided by $m$. Euler's Theorem: If $\text{gcd}(a, m) = 1$, then $a^{\phi(m)} \equiv 1 \pmod{m}$, where $\phi(m)$ is Euler's totient function. $\phi(m) = m \prod_{p|m, p \text{ prime}} (1 - \frac{1}{p})$. For $m=pq$ (distinct primes), $\phi(m) = (p-1)(q-1)$. RSA Cryptosystem Steps: Choose large primes $p, q$. Compute $n = pq$. Keep $p, q$ secret. Compute $\phi(n) = (p-1)(q-1)$. Choose $e$ such that $1 Public key: $(n, e)$. Compute deciphering key $d$ such that $ed \equiv 1 \pmod{\phi(n)}$, with $1 Encryption: Message $M \to C \equiv M^e \pmod{n}$. Decryption: Ciphertext $C \to M \equiv C^d \pmod{n}$. Binary Numbers Definition: Base-2 numbers using only digits $0$ and $1$. Binary to Decimal: For $N = (x_n x_{n-1} \dots x_1 x_0)_2$, $N = x_0 2^0 + x_1 2^1 + \dots + x_n 2^n$. Decimal to Binary: Repeatedly divide by 2 and record remainders. Complex Numbers Definition: $i$ is the imaginary unit, where $i^2 = -1$. $i^{4n} = 1$ $i^{4n+1} = i$ $i^{4n+2} = -1$ $i^{4n+3} = -i$ Complex Number Form: $z = a + bi$, where $a, b \in \mathbb{R}$. $\text{Re}(z) = a$ (real part) $\text{Im}(z) = b$ (imaginary part) Operations: Addition: $(a_1 + b_1 i) + (a_2 + b_2 i) = (a_1 + a_2) + (b_1 + b_2)i$ Multiplication: $(a_1 + b_1 i)(a_2 + b_2 i) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1)i$ Complex Conjugate: $\bar{z} = a - bi$. $\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$ $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$ $z \bar{z} = a^2 + b^2$ Modulus (Absolute Value): $|z| = \sqrt{a^2 + b^2} = \sqrt{z \bar{z}}$. $|z| \ge 0$, $|z|=0 \iff z=0$. $|z_1 z_2| = |z_1| |z_2|$. Triangle Inequality: $|z_1 + z_2| \le |z_1| + |z_2|$. Division: $\frac{z_1}{z_2} = \frac{z_1 \bar{z_2}}{|z_2|^2}$. Polar Form (Euler Formula): $z = |z|(\cos \theta + i \sin \theta) = |z|e^{i\theta}$, where $\theta$ is the argument. Indices (Exponents) Definition: $a^n = a \cdot a \cdot \dots \cdot a$ ($n$ times). Rules: $a^0 = 1$ (for $a \ne 0$) $a^{-n} = \frac{1}{a^n}$ $a^{1/n} = \sqrt[n]{a}$ $a^m a^n = a^{m+n}$ $\frac{a^m}{a^n} = a^{m-n}$ $(a^m)^n = a^{mn}$ $(ab)^m = a^m b^m$ $(\frac{a}{b})^m = \frac{a^m}{b^m}$ Logarithms Definition: If $N = b^x$, then $\log_b N = x$. (Base $b > 0, b \ne 1$) Special Cases: $\log_b b = 1$ $\log_b 1 = 0$ $\log_b (b^n) = n$ $b^{\log_b a} = a$ Types: Natural Logarithm: $\ln x = \log_e x$ (base $e \approx 2.718$) Common Logarithm: $\log x = \log_{10} x$ (base $10$) Properties: $\log_b (MN) = \log_b M + \log_b N$ $\log_b (M^n) = n \log_b M$ $\log_b (\frac{M}{N}) = \log_b M - \log_b N$ Change of Base: $\log_{b_2} N = \frac{\log_{b_1} N}{\log_{b_1} b_2}$ Antilogarithm: If $\log_b y = x$, then $y = \text{antilog}_b(x) = b^x$. Real-Life Applications (Examples) Richter Scale for Earthquakes: $R = \log_{10} (\frac{\lambda}{\lambda_0})$ Population Growth/Decay: Exponential models, often involving logarithms to solve for time. Compound Interest: $A = P(1 + r/n)^{nt}$ (exponential growth) Set Theory Basics Set: A well-defined collection of distinct objects. Elements/Members: Objects belonging to a set. Notation: $a \in A$ (a belongs to A), $b \notin A$ (b does not belong to A). Representation: Roster/Tabular Form: List all elements $\{1, 2, 3\}$. Set-Builder Form: Describe properties $\{x : x \text{ is a natural number and } x Special Sets: $\mathbb{N}$: Natural numbers $\{1, 2, 3, \dots\}$ $\mathbb{Z}$: Integers $\{\dots, -1, 0, 1, \dots\}$ $\mathbb{W}$: Whole numbers $\{0, 1, 2, \dots\}$ $\mathbb{Q}$: Rational numbers $\mathbb{R}$: Real numbers Types of Sets Singleton Set: Contains only one element. E.g., $\{2\}$. Empty Set ($\emptyset$ or $\{\}$): Contains no elements. Cardinality $n(\emptyset)=0$. Finite Set: Has a definite (countable) number of elements. $n(A)$ is its cardinality. Infinite Set: Not finite. Elements cannot be numbered $1$ to $n$. Equal Sets: $A=B$ if they have exactly the same elements. Order and repetition don't matter. Subset ($A \subseteq B$): Every element of $A$ is also an element of $B$. $\emptyset$ is a subset of every set. Every set is a subset of itself. Proper Subset ($A \subset B$): $A \subseteq B$ and $A \ne B$. Power Set ($P(A)$): The set of all subsets of $A$. If $n(A)=n$, then $n(P(A))=2^n$. Universal Set ($U$): A set containing all elements relevant to a particular context. Intervals (Subsets of $\mathbb{R}$) Closed Interval: $[a, b] = \{x : a \le x \le b\}$. Includes endpoints. Open Interval: $(a, b) = \{x : a Semi-Open/Closed: $[a, b) = \{x : a \le x $(a, b] = \{x : a Infinite Intervals: $(a, \infty) = \{x : x > a\}$ $[a, \infty) = \{x : x \ge a\}$ $(-\infty, b) = \{x : x $(-\infty, b] = \{x : x \le b\}$ $(-\infty, \infty) = \mathbb{R}$ Length of Interval: $b-a$. Venn Diagrams Pictorial representation of set relationships. Universal set $U$ is a rectangle, subsets are circles. Operations on Sets Operation Symbol Definition Properties Union $A \cup B$ $\{x : x \in A \text{ or } x \in B\}$ $A \cup B = B \cup A$ (Commutative) $(A \cup B) \cup C = A \cup (B \cup C)$ (Associative) $A \cup \emptyset = A$ $A \cup A = A$ $U \cup A = U$ Intersection $A \cap B$ $\{x : x \in A \text{ and } x \in B\}$ $A \cap B = B \cap A$ (Commutative) $\emptyset \cap A = \emptyset$, $U \cap A = A$ $A \cap A = A$ $(A \cap B) \cap C = A \cap (B \cap C)$ (Associative) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ (Distributive) Difference $A - B$ $\{x : x \in A \text{ and } x \notin B\}$ $A - B \ne B - A$ $(A-B) \cup (B-A) \cup (A \cap B) = A \cup B$ $A-B, B-A, A \cap B$ are disjoint. Complement $A'$ or $A^c$ $\{x : x \in U \text{ and } x \notin A\}$ $(A')' = A$ $A \cup A' = U$, $A \cap A' = \emptyset$ De Morgan's Laws: $(A \cup B)' = A' \cap B'$ $(A \cap B)' = A' \cup B'$ $\emptyset' = U$, $U' = \emptyset$ Practical Problems on Sets (Cardinality) $n(A \cup B) = n(A) + n(B)$ if $A \cap B = \emptyset$. $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ (general case). $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$. $n(A \text{ only}) = n(A - B) = n(A) - n(A \cap B)$. Relations Ordered Pair: $(a, b)$, where order matters. $(a, b) = (c, d) \iff a=c \text{ and } b=d$. Cartesian Product ($A \times B$): Set of all ordered pairs $(a, b)$ where $a \in A, b \in B$. $n(A \times B) = n(A) \times n(B)$. $A \times B \ne B \times A$ in general. Relation ($R$): A subset of $A \times B$. If $(a, b) \in R$, we write $a R b$. Domain of $R$: Set of all first elements of ordered pairs in $R$. Range of $R$: Set of all second elements of ordered pairs in $R$. Co-domain of $R$: The set $B$. (Range $\subseteq$ Co-domain). Number of Relations: If $n(A)=p, n(B)=q$, total relations from $A$ to $B$ are $2^{pq}$. Types of Relations Let $R$ be a relation on a set $A$ (i.e., $R \subseteq A \times A$). Empty Relation: $R = \emptyset \subset A \times A$. No element is related to any element. Universal Relation: $R = A \times A$. Each element is related to every element. Reflexive: $(a, a) \in R$ for all $a \in A$. (Every element is related to itself). Symmetric: If $(a, b) \in R$, then $(b, a) \in R$ for all $a, b \in A$. Transitive: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$ for all $a, b, c \in A$. Equivalence Relation: A relation that is reflexive, symmetric, and transitive. Functions Definition: A function $f: A \to B$ is a special type of relation from $A$ to $B$ that assigns to every element in $A$ a unique image in $B$. Domain of $f$: Set $A$. Range of $f$: Set of all images in $B$. (Range $\subseteq$ Co-domain $B$). A relation is a function if: Every element in the domain $A$ is mapped to an element in $B$. Each element in $A$ is mapped to exactly one element in $B$. Sequences and Series Sequence: An ordered list of numbers $a_1, a_2, a_3, \dots$. Finite Sequence: Has a countable number of terms. Infinite Sequence: Has an uncountable number of terms. Term: Each number in a sequence, denoted $a_n$ for the $n$-th term. Series: The sum of the terms of a sequence, $a_1 + a_2 + a_3 + \dots$. Arithmetic Progression (AP) Definition: A sequence where the difference between consecutive terms is constant. $a_{n+1} - a_n = d$. First Term: $a_1$ (often denoted $a$). Common Difference: $d$. $n$-th Term: $a_n = a + (n-1)d$. Sum of First $n$ Terms: $S_n = \frac{n}{2}[2a + (n-1)d]$ or $S_n = \frac{n}{2}[a + l]$, where $l$ is the last term. Property: $S_n - S_{n-1} = a_n$. Arithmetic Mean (AM) of $a, b$: $A = \frac{a+b}{2}$. Inserting $k$ AMs: If $A_1, \dots, A_k$ are $k$ AMs between $a$ and $b$, then $a, A_1, \dots, A_k, b$ form an AP. The common difference is $d = \frac{b-a}{k+1}$. Selection of Terms: 3 terms: $a-d, a, a+d$ 4 terms: $a-3d, a-d, a+d, a+3d$ 5 terms: $a-2d, a-d, a, a+d, a+2d$ Geometric Progression (GP) Definition: A sequence where the ratio between consecutive terms is constant. $\frac{a_{k+1}}{a_k} = r$. First Term: $a_1$ (often denoted $a$). Common Ratio: $r$. $n$-th Term: $a_n = ar^{n-1}$. Sum of First $n$ Terms: $S_n = \frac{a(r^n - 1)}{r-1}$ (for $r \ne 1$) or $S_n = na$ (for $r=1$). Geometric Mean (GM) of $a, b$: $G = \sqrt{ab}$. Inserting $k$ GMs: If $G_1, \dots, G_k$ are $k$ GMs between $a$ and $b$, then $a, G_1, \dots, G_k, b$ form a GP. The common ratio is $r = (\frac{b}{a})^{\frac{1}{k+1}}$. Selection of Terms: 3 terms: $\frac{a}{r}, a, ar$ 4 terms: $\frac{a}{r^3}, \frac{a}{r}, ar, ar^3$ 5 terms: $\frac{a}{r^2}, \frac{a}{r}, a, ar, ar^2$ Relation between AM and GM For positive real numbers $a, b$: $AM \ge GM$. That is, $\frac{a+b}{2} \ge \sqrt{ab}$. Equality holds if $a=b$. Infinite Geometric Progression Sum to Infinity: For a GP with $|r| Permutations and Combinations Factorial: $n! = 1 \times 2 \times \dots \times n$. $0! = 1$. $n! = n \times (n-1)!$. Fundamental Principles of Counting Rule of Sum (Addition): If event E can occur in $m$ ways AND event F can occur in $n$ ways, and they cannot occur simultaneously, then E OR F can occur in $m+n$ ways. Rule of Product (Multiplication): If event E can occur in $m$ ways, FOLLOWED BY event F in $n$ ways, then E AND F can occur in $m \times n$ ways. Permutations (Order Matters) $n$ distinct objects taken $r$ at a time (no repetition): $P(n, r) = \frac{n!}{(n-r)!}$. $n$ distinct objects taken $r$ at a time (with repetition): $n^r$. $n$ objects with repetitions ($p_1$ of type 1, $p_2$ of type 2, etc.): $\frac{n!}{p_1! p_2! \dots p_k!}$. Circular Permutations $n$ distinct objects in a circle: $(n-1)!$. $n$ distinct objects in a circle (if clockwise/anticlockwise are indistinguishable, e.g., necklace): $\frac{(n-1)!}{2}$. Combinations (Order Does Not Matter) $n$ distinct objects taken $r$ at a time (no repetition): $C(n, r) = \frac{n!}{r!(n-r)!}$. Properties: $C(n, 0) = 1$, $C(n, n) = 1$. $C(n, r) = C(n, n-r)$. $C(n, r) + C(n, r-1) = C(n+1, r)$. If $C(n, a) = C(n, b)$, then $a=b$ or $n=a+b$. $n$ objects taken $r$ at a time (with repetition): $C(n+r-1, r)$. Calculus - Introduction Variables: Quantities that change. Independent (input) and Dependent (output). Function: A rule that defines a relationship where each input (independent variable) has a unique output (dependent variable). Notation: $y = f(x)$. Representation: Equation, graph, table. Vertical Line Test: A curve is a function if no vertical line intersects it more than once. Function Operations Addition: $(f+g)(x) = f(x) + g(x)$. Subtraction: $(f-g)(x) = f(x) - g(x)$. Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$. Division: $(f/g)(x) = f(x)/g(x)$, provided $g(x) \ne 0$. Domain and Range Domain: Set of all possible input values for which the function is defined. Range: Set of all possible output values of the function. Types of Functions Polynomial Function: $f(x) = a_n x^n + \dots + a_1 x + a_0$. (Coefficients $a_i \in \mathbb{R}$, $n \in \mathbb{N}_0$). Rational Function: $f(x) = \frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are polynomials and $Q(x) \ne 0$. Exponential Function: $f(x) = a^x$, where $a > 0, a \ne 1$. Domain: $\mathbb{R}$, Range: $(0, \infty)$. If $a>1$, increasing; if $0 Logarithmic Function: $y = \log_a x \iff x = a^y$, where $a > 0, a \ne 1$. (Inverse of exponential). Greatest Integer Function (Floor Function): $f(x) = \lfloor x \rfloor$, the greatest integer less than or equal to $x$. (Step function). Modulus Function (Absolute Value): $f(x) = |x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x Signum Function: $\text{sgn}(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x Limits Intuitive Definition: The value a function "approaches" as the input "approaches" some value. Notation: $\lim_{x \to a} f(x) = L$. One-Sided Limits: Left-hand limit: $\lim_{x \to a^-} f(x)$. Right-hand limit: $\lim_{x \to a^+} f(x)$. $\lim_{x \to a} f(x)$ exists if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$. Algebra of Limits If $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist: $\lim_{x \to a} (f(x) \pm g(x)) = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$. $\lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$. $\lim_{x \to a} (\frac{f(x)}{g(x)}) = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \ne 0$. Continuity of a Function A function $f(x)$ is continuous at $x=a$ if: $f(a)$ is defined. $\lim_{x \to a} f(x)$ exists. $\lim_{x \to a} f(x) = f(a)$. Intuitively, a function is continuous if its graph can be drawn without lifting the pen. Derivatives (Instantaneous Rate of Change) Definition: The derivative of $f(x)$ with respect to $x$, denoted $f'(x)$ or $\frac{dy}{dx}$, measures the instantaneous rate of change of $y$ with respect to $x$. Geometric Interpretation: The slope of the tangent line to the graph of $f(x)$ at a given point. Definition using Limits (First Principle): $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Derivative Rules Function Derivative $c$ (constant) $0$ $x^n$ $nx^{n-1}$ $cf(x)$ $cf'(x)$ $f(x) \pm g(x)$ $f'(x) \pm g'(x)$ Product Rule: $f(x)g(x)$ $f'(x)g(x) + f(x)g'(x)$ Quotient Rule: $\frac{f(x)}{g(x)}$ $\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ $e^x$ $e^x$ $a^x$ $a^x \ln a$ $\ln x$ $\frac{1}{x}$ $\log_a x$ $\frac{1}{x \ln a}$ $\sin x$ $\cos x$ $\cos x$ $-\sin x$ $\tan x$ $\sec^2 x$ Chain Rule For a composite function $f(g(x))$: $\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$. Tangent Line and Equation of Tangent The derivative $f'(a)$ gives the slope of the tangent line to the curve $y=f(x)$ at the point $x=a$. Steps to find tangent line equation at $(a, f(a))$: Find $f'(x)$. Calculate slope $m = f'(a)$. Calculate $y$-coordinate $b = f(a)$. Use point-slope form: $y - b = m(x - a)$ or $y = m(x - a) + b$. Probability Random Experiment: Has more than one possible outcome, and outcome cannot be predicted in advance. Outcome: A result of a random experiment. Sample Space ($S$): The set of all possible outcomes. Trial: A particular performance of a random experiment. Event ($E$): A subset of the sample space. Impossible Event: $E = \emptyset$, $P(E)=0$. Sure Event: $E = S$, $P(E)=1$. Probability of an Event: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. (For equally likely outcomes). Complementary Event: $E'$ (not E). $P(E') = 1 - P(E)$. Mutually Exclusive Events: $E \cap F = \emptyset$. Occurrence of one precludes occurrence of the other. Exhaustive Events: A set of events whose union is the entire sample space ($E_1 \cup E_2 \cup \dots \cup E_n = S$). Independent Events: Occurrence of one does not affect the probability of the other. $P(E \cap F) = P(E)P(F)$. Dependent Events: Not independent. Conditional Probability Definition: The probability of event $E$ occurring, given that event $F$ has already occurred. Notation: $P(E|F)$. Formula: $P(E|F) = \frac{P(E \cap F)}{P(F)}$, provided $P(F) \ne 0$. Multiplication Rule for Probabilities: $P(E \cap F) = P(F)P(E|F)$. If $E, F$ are independent, $P(E \cap F) = P(E)P(F)$. Theorem of Total Probability If $A_1, A_2, \dots, A_n$ form a partition of the sample space $S$ (i.e., $A_i$ are mutually exclusive, exhaustive, and non-empty), and $B$ is any event, then: $P(B) = P(B|A_1)P(A_1) + P(B|A_2)P(A_2) + \dots + P(B|A_n)P(A_n)$. Bayes' Theorem If $A_1, A_2, \dots, A_n$ form a partition of the sample space $S$, and $B$ is any event with non-zero probability, then for any $A_i$: $P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_{j=1}^n P(B|A_j)P(A_j)}$. Prior Probability: $P(A_i)$ (probability before new evidence). Posterior Probability: $P(A_i|B)$ (revised probability after new evidence $B$). Straight Lines Inclination ($\theta$): Angle a line makes with the positive x-axis ($0^\circ \le \theta Slope ($m$): $m = \tan \theta$. Slope of line through $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Parallel Lines: $m_1 = m_2$. Perpendicular Lines: $m_1 m_2 = -1$. Angle ($\alpha$) between two lines with slopes $m_1, m_2$: $\tan \alpha = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$. Forms of Line Equations x-axis: $y=0$. y-axis: $x=0$. Parallel to y-axis: $x=a$. Parallel to x-axis: $y=b$. Point-Slope Form: $y - y_1 = m(x - x_1)$. (Slope $m$, through $(x_1, y_1)$). Two-Point Form: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$. (Through $(x_1, y_1)$ and $(x_2, y_2)$). Slope-Intercept Form: $y = mx + c$. (Slope $m$, y-intercept $c$). Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$. (x-intercept $a$, y-intercept $b$). Normal Form: $x \cos \alpha + y \sin \alpha = p$. ($p$ is perpendicular distance from origin, $\alpha$ is angle of normal with x-axis). General Form: $Ax + By + C = 0$. (Represents a line if $A, B$ not both zero). Distance Formulas Perpendicular distance from $(x_1, y_1)$ to $Ax + By + C = 0$: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$. Distance between parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$: $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$. Circles Definition: Set of all points equidistant from a fixed point (center). Radius: Constant distance $r$. Forms of Circle Equations Standard Form (center at origin $(0,0)$): $x^2 + y^2 = r^2$. Central Form (center at $(h,k)$): $(x-h)^2 + (y-k)^2 = r^2$. Diameter Form (endpoints of diameter $(x_1, y_1), (x_2, y_2)$): $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$. General Form: $x^2 + y^2 + 2gx + 2fy + c = 0$. Center: $(-g, -f)$. Radius: $r = \sqrt{g^2 + f^2 - c}$. Condition for a real circle: $g^2 + f^2 - c > 0$. If $g^2 + f^2 - c = 0$, it's a point circle. If $g^2 + f^2 - c Properties of General Form: No $xy$ term. Coefficients of $x^2$ and $y^2$ are equal (and usually 1). For $ax^2 + ay^2 + 2gx + 2fy + c = 0$, center is $(-\frac{g}{a}, -\frac{f}{a})$ and radius is $\frac{\sqrt{g^2 + f^2 - ac}}{|a|}$. Parabola Definition: Set of all points equidistant from a fixed line (directrix) and a fixed point (focus not on the line). Axis: Line through focus, perpendicular to directrix. Vertex: Point where parabola intersects its axis. Latus Rectum: Chord through focus, perpendicular to axis. Length is $4a$. Standard Forms of Parabola Equations (Vertex at origin $(0,0)$) Opens Right: $y^2 = 4ax$. Focus: $(a, 0)$. Directrix: $x = -a$. Axis: x-axis ($y=0$). Opens Left: $y^2 = -4ax$. Focus: $(-a, 0)$. Directrix: $x = a$. Axis: x-axis ($y=0$). Opens Up: $x^2 = 4ay$. Focus: $(0, a)$. Directrix: $y = -a$. Axis: y-axis ($x=0$). Opens Down: $x^2 = -4ay$. Focus: $(0, -a)$. Directrix: $y = a$. Axis: y-axis ($x=0$). In all cases, $a>0$.