5) Second order linear differential equations

Second order linear differential equations involve the second derivative of the unknown function and can be written in the form y'' + p(x)y' + q(x)y = r(x)
Homogeneous second order linear differential equations with constant coefficients have characteristic equations that can be used to find their solutions
Non-homogeneous second order linear differential equations can be solved using the method of undetermined coefficients or the method of variation of parameters
Particular integrals can be found for specific types of non-homogeneous functions, such as eax, sin ax, cos ax, ax^2 + bx + c (a, b, c are real numbers)
Second order linear differential equations are used to model many physical phenomena, such as oscillations, waves, and electromagnetic fields.

Important Formulas to Remember in these topic

Explanation and solution methods for second order differential equations with constant coefficients, including:

- Homogeneous equations, such as:

$$\frac{d^2y}{dx^2} + ay = 0 \Rightarrow y = c_1\cos(\sqrt{a}x) +$c_2\sin(\sqrt{a}x)$$

where a is a constant and c_1 and c_2 are arbitrary constants.

- Non-homogeneous equations, such as:

$$\frac{d^2y}{dx^2} + ay = f(x)$$

where f(x) is a known function. One solution can be found by the method of undetermined coefficients, which involves finding a particular solution based on the form of f(x). The general solution can then be found by adding the homogeneous solution to the particular solution.

- Complex roots, where the homogeneous equation has the form:

$$\frac{d^2y}{dx^2} + a_1\frac{dy}{dx} + a_2y = 0$$

and the roots of the characteristic equation are complex, such as:

$$r_1 = -\alpha + i\beta, \quad r_2 = -\alpha - i\beta$$

The general solution can be written as:

$$y = e^{-\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$$

where alpha and beta are constants, and c_1 and c_2 are arbitrary constants determined by initial or boundary conditions.

4) First order differential equations

First order differential equations involve only the first derivative of the unknown function
Variable separable equations involve separating the variables in the equation and integrating each side to obtain the solution
Homogeneous equations involve replacing the function with a new variable that simplifies the equation, and then solving it using separation of variables
Exact equations involve finding an integrating factor that makes the equation exact, and then integrating to obtain the solution
Bernoulli's equation involves transforming a nonlinear first order equation into a linear equation using a substitution, and then solving it using an integrating factor

Important Formulas to Remember in these topic

Explanation and solution methods for first order differential equations, including:

- Variable separable equations, such as:

$$\frac{dy}{dx} = f(x)g(y) \Rightarrow \int\frac{1}{g(y)}dy = \int f(x)dx + C$$

- Homogeneous equations, such as:

$$\frac{dy}{dx} = f\left(\frac{y}{x}\right) \Rightarrow y = vx \Rightarrow \frac{dv}{dx} = \frac{v-f(v)}{x}$$

- Exact equations, such as:

$$M(x,y)dx + N(x,y)dy = 0 \Rightarrow \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \Rightarrow \text{solution } F(x,y) = C$$

- Linear differential equation of the form dy/dx+Py=Q, such as:

$$\frac{dy}{dx} + P(x)y = Q(x) \Rightarrow y = e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx + C\right)$$

- Bernoulli's equation, such as:

$$\frac{dy}{dx} + P(x)y = Q(x)y^n \Rightarrow z = y^{1-n} \Rightarrow \frac{dz}{dx} + (1-n)P(x)z = (1-n)Q(x)$$

3) Formation of differential equations

Differential equations can be formed using various methods, including physical principles, geometry, and other mathematical models
The process of forming a differential equation from a given situation involves identifying the relevant variables, determining how they are related, and representing this relationship as an equation involving derivatives
In some cases, the boundary conditions or initial conditions must also be considered to obtain a unique solution
Differential equations can be classified according to their properties, such as linearity and order, which can provide insight into their solutions
The ability to form differential equations is an important skill for scientists and engineers who need to model complex systems

Important Formulas to Remember in these topic

There are various methods for creating differential equations, including:

- Direct integration, such as:

$$\frac{dy}{dx} = kx \Rightarrow y = \frac{kx^2}{2} + C$$

- Separation of variables, such as:

$$\frac{dy}{dx} = \frac{y}{x} \Rightarrow \frac{dy}{y} = \frac{dx}{x} \Rightarrow \ln|y| = \ln|x| + C$$

2) Order and degree of a differential equation

The order of a differential equation is the highest order derivative in the equation
The degree of a differential equation is the highest power of the highest order derivative in the equation
The order and degree of a differential equation can affect the complexity of its solution
Most commonly encountered differential equations are first or second order, but higher order equations are also possible
It is important to correctly identify the order and degree of a differential equation before attempting to solve it

Important Formulas to Remember in these topic

The order of a differential equation is the highest derivative that appears in the equation, such as:

$$\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$$

The degree of a differential equation is the power to which the highest derivative is raised, after the equation has been written in standard form, such as:

$$\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + y = 0$$

1) Definition of a differential equation

A differential equation is an equation that involves an unknown function and its derivatives
Differential equations are used to model real-world phenomena in many fields, including physics, engineering, economics, and biology
A differential equation is called an ordinary differential equation (ODE) if it involves only one independent variable, and a partial differential equation (PDE) if it involves multiple independent variables
Solutions to differential equations can be found using various techniques, including separation of variables, integrating factors, and Laplace transforms
Differential equations are an important topic in mathematics and have many practical applications

Important Formulas to Remember in these topic

A differential equation is an equation that relates a function and its derivatives to one another, such as:

$$\frac{dy}{dx} = f(x,y)$$

8) Mean and RMS values, Trapezoidal rule and Simpson’s 1/3 Rule for approximation integrals:

The mean value of a function over an interval is the integral of the function divided by the length of the interval.
The RMS value of a function over an interval is the square root of the integral of the square of the function divided by the length of the interval.
The trapezoidal rule approximates the area under a curve by dividing the area into trapezoids.
Simpson's 1/3 rule approximates the area under a curve by dividing the area into parabolic segments

Important Formulas to Remember in these topic

$$\text{Mean value } = \frac{1}{b-a}\int_a^b f(x)dx$$

$$\text{RMS value } = \sqrt{\frac{1}{b-a}\int_a^b [f(x)]^2 dx}$$

$$\text{Trapezoidal rule } = \frac{b-a}{2n}[f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n)]$$

$$\text{Simpson's 1/3 Rule } = \frac{b-a}{6}\left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \dots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)\right]$$

7) Application of Integration to find areas under plane curves and volumes of Solids of revolution

Integration can be used to find the area under a curve, which can represent the distance traveled by an object or the work done by a force.
It can also be used to find the volume of a solid of revolution, which is formed by rotating a curve around an axis.

Important Formulas to Remember in these topic

$$\text{Area } = \int_a^b f(x)dx$$

$$\text{Volume } = \int_a^b \pi y^2 dx$$

$$\text{Volume } = \int_a^b \pi R^2 dx$$

6) Definite Integrals and properties, Definite Integral as the limit of a sum:

A definite integral is the area under a curve between two endpoints.
It can be calculated using the limit of a sum of rectangles that approximate the area under the curve.
The properties of definite integrals include linearity, additivity, and symmetry.

Important Formulas to Remember in these topic

$$\int_a^b f(x)dx = F(b) - F(a)$$

$$\int_a^b f(x)dx = -\int_b^a f(x)dx$$

$$\int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx$$

$$\int_a^b kf(x)dx = k\int_a^b f(x)dx$$

$$\int_a^b f(x)dx + \int_b^c f(x)dx = \int_a^c f(x)dx$$

$$\int_a^b f(x)dx = \lim_{n\to\infty}\frac{b-a}{n}\sum_{i=1}^{n} f(x_i)$$

5) Integration by parts:

Integration by parts involves integrating the product of two functions.
The technique involves choosing one function to differentiate and the other to integrate.
The resulting integral involves an integral of the product of the two functions, which can often be simplified.

Important Formulas to Remember in these topic

$$\int u\frac{dv}{dx}dx = uv - \int v\frac{du}{dx}dx$$

4) Integration of reducible and irreducible quadratic factors:

A reducible quadratic factor can be factored into linear factors and then integrated using standard integration rules.
An irreducible quadratic factor cannot be factored and requires a special rule for integration.
For example, the integral of 1/(x^2 + a^2) is (1/a) arctan(x/a).

Important Formulas to Remember in these topic

$$\int \frac{Ax + B}{(ax^2 + bx + c)^n}dx$$

3) Integration by substitution

Integration by substitution involves substituting a new variable for the variable of integration.
This is useful when the integrand contains a function that can be simplified by substitution.
The technique involves choosing a new variable, computing its derivative, and substituting into the integral.
The resulting integral is often simpler to solve.

Important Formulas to Remember in these topic

$$\int f(g(x))g'(x)dx = \int f(u)du$$

2) Integration by decomposition of the integrand, integration of trigonometric, algebraic, exponential, logarithmic and Hyperbolic functions

Integration by decomposition involves breaking down a complicated function into simpler functions that can be integrated separately.
Trigonometric functions include sine, cosine, tangent, etc. and have their own integration rules.
Algebraic functions include polynomials and rational functions.
Exponential functions involve e^x or a^x, where a is a constant.
Logarithmic functions involve ln(x) or log(x), where x is a positive real number.
Hyperbolic functions include sinh, cosh, tanh, etc. and also have their own integration rules.

Important Formulas to Remember in these topic

$$\int f(x) dx = \int g(x)h(x) dx $$

$$\int \sin^m(x) \cos^n(x) dx $$

$$\int \frac{P(x)}{Q(x)} dx$$

1) Indefinite Integral – Standard forms

An indefinite integral is the anti-derivative of a function.
Standard forms of indefinite integrals include basic rules such as the power rule, the constant rule, and the sum rule.
The power rule states that the integral of x^n is (1/(n+1)) x^(n+1).
The constant rule states that the integral of a constant is the constant times the variable of integration.
The sum rule states that the integral of a sum of functions is the sum of the integrals of each individual function.

Important Formula to Remember in these topic

$$\int f(x) dx$$

Partial Differentiation - Partial derivatives up to second order - Euler's theorem

Partial Derivative Formulas

Partial Differentiation

$$\frac{\partial z}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h}$$

$$\frac{\partial z}{\partial y} = \lim_{k \to 0} \frac{f(x, y + k) - f(x, y)}{k}$$

Partial Derivatives up to Second Order

Second Order Partial Derivatives: $$\frac{\partial^2z}{\partial x^2} \quad \frac{\partial^2z}{\partial y^2} \quad \frac{\partial^2z}{\partial x \partial y} = \frac{\partial^2z}{\partial y \partial x}$$

Euler's Theorem

$$\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}\cdot\frac{dy}{dx} = 0$$

Geometrical applications of the derivative (angle between curves, tangent and normal) - Increasing and decreasing functions - Maxima and Minima (single variable functions) using second order derivative only physical application - Rate Measure

Geometrical applications of the derivative:

The derivative can be used to find the slope of a tangent line to a curve at a given point.
The derivative can also be used to find the equation of a normal line to a curve at a given point, which is a line perpendicular to the tangent line at that point.
The angle between two curves at a point of intersection can be found using the slopes of the tangent lines to each curve at that point.

Increasing and decreasing functions, maxima and minima:

A function is increasing if its derivative is positive and decreasing if its derivative is negative.
A function has a local maximum at a point where the derivative changes sign from positive to negative, and a local minimum at a point where the derivative changes sign from negative to positive.
A function has a global maximum or minimum at the highest or lowest point, respectively, on the entire domain of the function.
The second order derivative can be used to determine whether a local maximum or minimum is a maximum or minimum, respectively. Specifically, a local maximum is a maximum if the second order derivative is negative at that point, and a local minimum is a minimum if the second order derivative is positive.

Physical applications of the derivative:

The derivative can be used to measure the rate of change of a physical quantity, such as velocity, acceleration, or temperature.
The derivative of displacement with respect to time gives velocity, the derivative of velocity with respect to time gives acceleration, and so on.
The integral of the derivative can be used to find the total change in a physical quantity over a given interval, such as the total distance traveled by an object or the total heat energy absorbed by a material.

Important Formulas to Remember in these topic

Derivative Formulas

Geometrical Applications of the Derivative

Angle between Two Curves: $$\tan\theta = \frac{dy_1/dx - dy_2/dx}{1 + dy_1/dx\cdot dy_2/dx}$$
Tangent and Normal to a Curve: $$y - y_0 = \frac{dy}{dx}(x - x_0) \quad \text{(Tangent)}$$ $$y - y_0 = -\frac{dx}{dy}(x - x_0) \quad \text{(Normal)}$$

Derivative of a function with respect to another function - Second order derivatives

Derivatives with respect to another function:

When taking the derivative of a function with respect to another function, we use the chain rule, treating the outer function as the variable and the inner function as the function.
This process is also known as implicit differentiation, and is often used to find the slope of a curve at a specific point or to solve equations where the dependent variable is not explicitly given in terms of the independent variable.
For example, if we have y = f(g(x)), then we can find dy/dx by applying the chain rule: dy/dx = (dy/dg) * (dg/dx).

Second order derivatives:

The second order derivative of a function is the derivative of its first order derivative. It measures how the rate of change of a function changes as the input value changes.
The second order derivative is denoted by f''(x) or d^2y/dx^2, and is calculated by taking the derivative of the first order derivative of the function.
If the second order derivative is positive at a certain point, then the function is said to be concave up at that point, whereas if it is negative, the function is said to be concave down.
A point where the second order derivative is zero is known as an inflection point, where the function transitions from being concave up to concave down, or vice versa.
Second order derivatives can be used to find the maximum or minimum points of a function, as a maximum or minimum occurs at a point where the second order derivative changes sign.
In some cases, the second order derivative can be used to determine the behavior of the function at extreme values, such as asymptotes or singularities.
The second order derivative can be extended to functions of multiple variables, where it measures the rate of change of the gradient of the function.

Important Formulas to Remember in these topic

Derivative Formulas

Derivative of a Function with Respect to Another Function

$$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx} = f'(u)\cdot g'(x)$$

Second Order Derivatives

Second Derivative of a Function: $$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)$$

Product Rule for Second Derivatives: $$\frac{d^2}{dx^2}(f(x)g(x)) = f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)$$

Chain Rule for Second Derivatives: $$\frac{d^2y}{dx^2} = \frac{d^2y}{du^2}\cdot\left(\frac{du}{dx}\right)^2 + \frac{dy}{du}\cdot\frac{d^2u}{dx^2}$$

Differentiation: sum, product, quotient, trigonometric, inverse trigonometric, exponential, logarithmic, hyperbolic, implicit, explicit, and parametric functions.

The sum rule states that the derivative of the sum of two functions is equal to the sum of their derivatives.
The product rule states that the derivative of the product of two functions is equal to the first function times the derivative of the second, plus the second function times the derivative of the first.
The quotient rule states that the derivative of the quotient of two functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
The chain rule, also known as function of function rule, is used to differentiate composite functions, or functions of functions, and states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
The derivative of trigonometric functions depends on the specific trigonometric function, but can generally be found using basic trigonometric identities.
The derivative of inverse trigonometric functions is found using differentiation by substitution and the inverse function theorem.
The derivative of exponential functions is equal to the function itself, multiplied by the natural logarithm of the base e.
The derivative of logarithmic functions is found using the logarithmic differentiation technique, which involves taking the natural logarithm of both sides of the equation and differentiating using the chain rule.
The derivative of hyperbolic functions is similar to that of trigonometric functions and can be found using basic identities.
The differentiation of implicit, explicit, and parametric functions involves different techniques, but typically involves applying the chain rule and solving for the derivative.

Important Formulas to Remember in these topic

Differentiation Formulas

Sum of Functions: $$\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$$

Product of Functions: $$\frac{d}{dx}(f(x)g(x)) = f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x)$$

Quotient of Functions: $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{g(x)^2}$$

Function of Function: $$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$$

Trigonometric Functions: $$\frac{d}{dx}\sin(x) = \cos(x), \quad \frac{d}{dx}\cos(x) = -\sin(x), \quad \frac{d}{dx}\tan(x) = \sec^2(x)$$

Inverse Trigonometric Functions: $$\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arccos(x) = -\frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arctan(x) = \frac{1}{1+x^2}$$

Exponential Functions: $$\frac{d}{dx}e^x = e^x$$

Logarithmic Functions: $$\frac{d}{dx}\ln(x) = \frac{1}{x}, \quad \frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$$

Hyperbolic Functions: $$\frac{d}{dx}\sinh(x) = \cosh(x), \quad \frac{d}{dx}\cosh(x) = \sinh(x), \quad \frac{d}{dx}\tanh(x) = \operatorname{sech}^2(x)$$

Implicit Functions: $$\frac{dy}{dx} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}$$

Explicit Functions: $$\frac{d}{dx}y = \frac{dy}{dx}$$

Parametric Functions: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

Functions and limits - Standard limits

Standard limits are specific values that help to determine the limits of functions as they approach certain values.
There are six standard limits, including the limit of a constant function, the limit of a linear function, the limit of a quadratic function, the limit of a square root function, the limit of a rational function, and the limit of an exponential function.
Knowing the standard limits can help you quickly determine the limit of more complex functions by applying them to smaller parts of the larger function.
The limit of a function is the value that the function approaches as the input approaches a certain value, and may or may not be equal to the actual value of the function at that point.
Understanding standard limits is a crucial part of calculus, as they form the foundation for more advanced topics such as derivatives and integrals.

Important Formulas to Remember in these Functions and limits

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

$$\lim_{x \to 0} \frac{\tan x}{x} = 1$$

$$\lim_{x \to 0} (1+x)^{\frac{1}{x}} = e$$

$$\lim_{x \to \infty} \left(1+\frac{1}{x}\right)^x = e$$

$$\lim_{x \to \infty} \left(1+\frac{k}{x}\right)^x = e^k$$

$$\lim_{x \to 0} \frac{e^x-1}{x} = 1$$

$$\lim_{x \to \infty} \frac{\ln x}{x} = 0$$

$$\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$$

$$\lim_{x \to \infty} \frac{a^x}{x!} = 0$$

$$\lim_{x \to \infty} \frac{x^k}{a^x} = 0$$

$$\lim_{x \to \infty} \sqrt[x]{x} = 1$$

Conic Section

Properties of parabola:

A parabola is a set of all points that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).
The axis of symmetry of a parabola is a line that passes through the focus and is perpendicular to the directrix.
The vertex of the parabola is the point where the axis of symmetry intersects the parabola.
The standard form of a parabola with vertex at the origin is y^2 = 4px, where p is the distance from the vertex to the focus.

Properties of ellipse:

An ellipse is a set of all points such that the sum of the distances from two fixed points (called foci) is constant.
The major axis of an ellipse is the longest diameter and contains the two foci. The minor axis is the perpendicular diameter.
The center of an ellipse is the midpoint of the major axis.
The standard form of an ellipse with the center at the origin is x^2/a^2 + y^2/b^2 = 1, where a and b are the lengths of the semi-major and semi-minor axes.

Properties of hyperbola:

A hyperbola is a set of all points such that the difference of the distances from two fixed points (called foci) is constant.
The transverse axis of a hyperbola is the longest diameter and contains the two vertices. The conjugate axis is the perpendicular diameter.
The center of a hyperbola is the midpoint of the transverse axis.
The standard form of a hyperbola with the center at the origin is x^2/a^2 - y^2/b^2 = 1, where a and b are the lengths of the semi-transverse and semi-conjugate axes.

Important Points to Remember in these Properties of parabola, ellipse and hyperbola:

A parabola is the set of all points that are equidistant from a fixed point (focus) and a fixed line (directrix).
An ellipse is the set of all points such that the sum of the distances from two fixed points (foci) is constant.
A hyperbola is the set of all points such that the difference of the distances from two fixed points (foci) is constant.

Standard forms with vertex at origin:

The standard form of a parabola with vertex at the origin is y^2 = 4px, where p is the distance from the vertex to the focus.
The standard form of an ellipse with the center at the origin is x^2/a^2 + y^2/b^2 = 1, where a and b are the lengths of the semi-major and semi-minor axes.
The standard form of a hyperbola with the center at the origin is x^2/a^2 - y^2/b^2 = 1, where a and b are the lengths of the semi-transverse and semi-conjugate axes.
For all three types of conic sections, if the vertex is not at the origin, the equations can be shifted by replacing x with (x - h) and y with (y - k), where (h, k) is the coordinates of the vertex.

Important Formulas to Remember in these Standard forms with vertex at origin:

Parabola: $$y^2 = 4px$$
Ellipse: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
Hyperbola: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

Additional properties of conic sections:

The eccentricity of an ellipse is the ratio of the distance between the foci to the length of the major axis. It is always less than 1.
The eccentricity of a hyperbola is the ratio of the distance between the foci to the length of the transverse axis. It is always greater than 1.
The directrix of a parabola is a line that is equidistant to the focus and perpendicular to the axis of symmetry.
The asymptotes of a hyp-erbola are two lines that the hyperbola approaches but never intersects. The slopes of the asymptotes are ±(b/a).

Circles

1) Equation of circle given center and radius:

The equation of a circle with center (a,b) and radius r is given by (x-a)² + (y-b)² = r².
This equation can be derived from the distance formula, by setting the distance between (x,y) and (a,b) equal to r.
The center of the circle is the point (a,b), and the radius is r.
The equation can be written in general form as x² + y² + Dx + Ey + F = 0, where D = -2a, E = -2b, and F = a² + b² - r².
The standard form of the equation is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle.

Important Formulas to Remember in these Equation of circle given center and radius:

Standard form: $$(x - a)^2 + (y - b)^2 = r^2$$

2) General equation:

The general equation of a circle is given by Ax² + Ay² + Bx + Cy + D = 0, where A and B are not both zero.
This equation can be obtained by substituting x² and y² with their respective coefficients in the standard form of the equation.
The coefficients A, B, and C determine the position of the circle in the xy-plane, while D determines its size.
To convert the general equation to standard form, complete the square for both x and y, and then combine the resulting expressions.
The standard form of the equation can be used to find the center and radius of the circle.

Important Formulas to Remember in these General equation:

General form: $$ax^2 + by^2 + 2gx + 2fy + c = 0$$

3) Finding center and radius:

To find the center and radius of a circle given its equation in standard form, rewrite the equation in the form (x-h)² + (y-k)² = r².
Complete the square for both x and y, and then rearrange the terms to isolate the center (h,k) and the radius r.
The center of the circle is (h,k), and the radius is r.
To find the center and radius of a circle given its equation in general form, convert the equation to standard form first, and then use the method described above.
If the equation of the circle is given in another form, such as center and a point on the circumference or 3 non-collinear points, use the appropriate method to find the center and radius.

Important Formulas to Remember in these Finding center and radius:

Center: $$(a,b)$$
Radius: $$r = \sqrt{(x - a)^2 + (y - b)^2}$$

4) Center and a point on the circumference:

To find the center and radius of a circle given its center (h,k) and a point on the circumference (x1,y1), use the distance formula.
The distance between the center and the point on the circumference is equal to the radius of the circle.
The center of the circle is the midpoint of the line segment joining (h,k) and (x1,y1), and the radius is the distance between the center and (x1,y1).
Alternatively, use the equation of the circle to solve for the radius, and then substitute the coordinates of the point on the circumference to solve for the center.
There are two possible circles that can pass through a given center and point on the circumference, one with a positive radius and one with a negative radius.

Important Formulas to Remember in these Center and a point on the circumference:

Center: $$(a,b)$$
Point on circumference: $$(x_1,y_1)$$
Equation: $$(x - a)^2 + (y - b)^2 = (x_1 - a)^2 + (y_1 - b)^2$$

5) 3 non-collinear points:

To find the center and radius of a circle given three non-collinear points (x1,y1), (x2,y2), and (x3,y3), use the circumcenter formula.
The circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle formed by the three points.
Find the slope and midpoint of each side of the triangle, and then find the equations of the perpendicular bisectors.
The point of intersection of the perpendicular bisectors is the center of the circle.
The radius of the circle is the distance between the center and any of the three points.

Important Formulas to Remember in these 3 non-collinear points:

Center: $$(a,b)$$
Points: $$(x_1,y_1), (x_2,y_2), (x_3,y_3)$$
Equation: $$\begin{aligned} a &= \frac{1}{2}\begin{vmatrix} x_1^2 + y_1^2 & y_1 & 1 \\ x_2^2 + y_2^2 & y_2 & 1 \\ x_3^2 + y_3^2 & y_3 & 1 \end{vmatrix} \\ b &= -\frac{1}{2}\begin{vmatrix} x_1^2 + y_1^2 & x_1 & 1 \\ x_2^2 + y_2^2 & x_2 & 1 \\ x_3^2 + y_3^2 & x_3 & 1 \end{vmatrix} \\ r &= \sqrt{(x_1 - a)^2 + (y_1 - b)^2} \end{aligned}$$

6) Center and tangent:

To find the center and radius of a circle given its center (h,k) and a tangent line at a point (x1,y1) on the circumference, use the perpendicular bisector of the tangent line.
The perpendicular bisector of the tangent line passes through the center of the circle.
Find the slope of the tangent line and the midpoint of the line segment joining (h,k) and (x1,y1), and then find the equation of the perpendicular bisector.
The point of intersection of the perpendicular bisector and the line passing through (h,k) and (x1,y1) is the center of the circle.
The radius of the circle is the distance between the center and (x1,y1).

Important Formulas to Remember in these Center and tangent:

Center: $$(a,b)$$
Point on tangent: $$(x_1,y_1)$$
Equation: $$(x - a)(x_1 - a) + (y - b)(y_1 - b) = r^2$$

7) Equation of tangent and normal at a point on the circle:

The tangent line to a circle at a point (x1,y1) on the circumference is perpendicular to the radius passing through the point.
The slope of the tangent line is equal to the negative reciprocal of the slope of the radius passing through the point.
The equation of the tangent line is y - y1 = m(x - x1), where m is the slope of the tangent line.
The normal line to the circle at the same point is perpendicular to the tangent line and passes through the point (x1,y1).
The slope of the normal line is equal to the negative of the slope of the tangent line, and the equation of the normal line is y - y1 = (-1/m)(x - x1), where m is the slope of the tangent line.

Important Formulas to Remember in these Equation of tangent and normal at a point on the circle:

Center: $$(a,b)$$
Point on circle: $$(x_1,y_1)$$
Equation of tangent: $$\frac{x(x_1 - a) + y(y_1 - b)}{r^2} = \frac{x_1 - a}{r} = \frac{y_1 - b}{r}$$
Equation of normal: $$\frac{x(a - x_1) + y(b - y_1)}{r^2} = \frac{a - x_1}{r} = \frac{b - y_1}{r}$$