Kelompok 1
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The document discusses circle equations with a given center point and radius. It provides the general equation for finding the distance between two points and uses this to derive the equation of a circle given its center P(-2,3) and passing through point Q(4,-1). It then shows the equations for circles with this center that meet exactly one point X and one point Y.
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Equation of a Circle in standard and general form
Writing an equation of a circle in standard or general form and finding the radius and the center of a circle given an equation.
General equation of a circle
- The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0.
- Comparing this equation to the standard form (x - h)2 + (y - k)2 = r2 allows us to determine the circle's center (h, k) and radius r from any circle equation.
- For the example circle x2 + y2 + 4x - 6y + 4 = 0, the center is (-2, 3) and the radius is 3 units.
Determining the center and the radius of a circle
This document provides instruction on determining the center and radius of a circle given its equation in standard form and vice versa. It begins with stating the objectives of identifying the standard form of a circle equation and using it to determine center and radius or write the equation given one of those. Several examples are worked through, including transforming equations to standard form and finding center and radius. Short exercises are provided for students to practice these skills.
Circle
1) The document provides information on circle equations and properties including: the general form of a circle equation, finding the center and radius from an equation, and determining if a point lies inside, outside or on a circle.
2) Examples are given for writing circle equations in different forms, finding centers and radii, and finding intersection points between circles and lines.
3) The key steps for finding intersections between a line and circle are outlined using simultaneous equations and the discriminant.
Nov. 3 Coordinate Equation Of A Circle
The document discusses the coordinate equation of a circle. It states that a circle can be defined as all points that are equidistant from the center point. The standard form of the equation of a circle is r2 = (x - h)2 + (y - k)2, where (h, k) are the coordinates of the center and r is the radius. It provides examples of writing the equation of a circle given properties like the center point and radius or a point on the circle.
003 circle
The document provides information about conic sections, specifically circles. It defines a circle as the set of points equidistant from a fixed point, and provides the standard equation (x - h)2 + (y - k)2 = r2, where (h, k) is the circle's center and r is the radius. Several example problems are worked through, finding the equation of circles given properties like specific points or tangency to lines. The concept of a family of circles is introduced, where the circles share a common property like center location. Radical axes are defined as the line perpendicular to the line joining two circle centers.
Tangents + intersections
- The document discusses finding the equation of a tangent line to a circle given a point of tangency, as well as finding points of intersection between a line and a circle.
- The key steps are to find the center and radius of the circle from its equation, then use properties of tangents (gradient of radius = -1/gradient of tangent) to determine the gradient of the tangent line.
- To find intersections, set the line and circle equations equal and solve using substitution or factorizing, looking for real number solutions. If only one solution is found, the line is tangent to the circle.
INTERACTIVE MULTIMEDIA on ARITHMETIC SEQUENCE_SIM FOR MATH GRADE 10.pdf
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Equation of a Circle in standard and general form
Writing an equation of a circle in standard or general form and finding the radius and the center of a circle given an equation.
General equation of a circle
- The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0.
- Comparing this equation to the standard form (x - h)2 + (y - k)2 = r2 allows us to determine the circle's center (h, k) and radius r from any circle equation.
- For the example circle x2 + y2 + 4x - 6y + 4 = 0, the center is (-2, 3) and the radius is 3 units.
Determining the center and the radius of a circle
This document provides instruction on determining the center and radius of a circle given its equation in standard form and vice versa. It begins with stating the objectives of identifying the standard form of a circle equation and using it to determine center and radius or write the equation given one of those. Several examples are worked through, including transforming equations to standard form and finding center and radius. Short exercises are provided for students to practice these skills.
Circle
1) The document provides information on circle equations and properties including: the general form of a circle equation, finding the center and radius from an equation, and determining if a point lies inside, outside or on a circle.
2) Examples are given for writing circle equations in different forms, finding centers and radii, and finding intersection points between circles and lines.
3) The key steps for finding intersections between a line and circle are outlined using simultaneous equations and the discriminant.
Nov. 3 Coordinate Equation Of A Circle
The document discusses the coordinate equation of a circle. It states that a circle can be defined as all points that are equidistant from the center point. The standard form of the equation of a circle is r2 = (x - h)2 + (y - k)2, where (h, k) are the coordinates of the center and r is the radius. It provides examples of writing the equation of a circle given properties like the center point and radius or a point on the circle.
003 circle
The document provides information about conic sections, specifically circles. It defines a circle as the set of points equidistant from a fixed point, and provides the standard equation (x - h)2 + (y - k)2 = r2, where (h, k) is the circle's center and r is the radius. Several example problems are worked through, finding the equation of circles given properties like specific points or tangency to lines. The concept of a family of circles is introduced, where the circles share a common property like center location. Radical axes are defined as the line perpendicular to the line joining two circle centers.
Tangents + intersections
- The document discusses finding the equation of a tangent line to a circle given a point of tangency, as well as finding points of intersection between a line and a circle.
- The key steps are to find the center and radius of the circle from its equation, then use properties of tangents (gradient of radius = -1/gradient of tangent) to determine the gradient of the tangent line.
- To find intersections, set the line and circle equations equal and solve using substitution or factorizing, looking for real number solutions. If only one solution is found, the line is tangent to the circle.
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The document provides examples and explanations of finding equations of circles that are tangent to lines or pass through given points. It discusses using formulas for perpendicular distance from a line or point to determine a circle's radius and center. It also shows solving systems of equations algebraically or using geometric constructions to find centers and radii when given additional constraints like points or lines that a circle must be tangent to or pass through.
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The document discusses the standard form of a circle equation and provides examples of writing the equation of various circles given their center and radius. It begins by defining the components of a circle - the center (h,k) and radius r. Then it shows the standard form of a circle equation: (x-h)2 + (y-k)2 = r2. The rest of the document provides the center and radius of several circles and has the reader write out the corresponding standard equation for each one.
Circle
This document discusses the general equation of a circle and provides examples of finding the center and radius of circles given their equations.
The general equation of a circle is given as (x - h)2 + (y - k)2 = r2, where (h,k) represents the center and r is the radius. Three examples are worked through, finding the center and radius for circles with equations x2 + y2 - 4x - 8y + 16 = 0, x2 + y2 - 10x - 20y + 100 = 0, and x2 + y2 - 2x + 1 = 5. Alternative methods for finding the center and radius are also discussed.
Circle
The document discusses circles and their equations. It defines a circle as all points in a plane that are a fixed distance from a fixed center point. This fixed distance is called the radius. The standard form of a circle equation is (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. It also discusses converting between the standard and general forms of a circle equation.
General equation of a circle
The document provides information about the standard equation of a circle given its center and tangency condition. It gives the equation (x – 5)2 + (y + 6)2 = 36 for the circle with center (5,-6) tangent to the x-axis. It then explains how to rewrite this in general form as x2 + y2 - 10x +12y+25 =0. It also gives the general form of the equation of a circle Ax2 + By2 + Cx + Dy + E = 0 and examples of identifying the center and radius from equations in general form.
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This document discusses graphing circles and converting between graphing form and standard form equations of circles. It provides examples of writing the equation of a circle given its center and radius in graphing form. It also shows how to find the center, radius, and equation of a circle given the standard form equation. The key points are:
- In graphing form, the center is denoted as (h, k) and the equation is (x - h)2 + (y - k)2 = r2
- To convert to standard form, complete the square and factor the left side, moving any constant to the right side.
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- Problem 3 discusses how the area between a circle and line is maximized when the line acts as the diameter.
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This document provides formulas and examples for calculating the distance and midpoint between two points.
The distance formula is used to find the distance between two points (x1, y1) and (x2, y2). The midpoint formula finds the midpoint between two points by taking the average of the x-coordinates and y-coordinates.
Several examples demonstrate using the distance and midpoint formulas to find distances, midpoints, and missing endpoint coordinates. The key steps are to work inside parentheses, do exponents before addition, and plug points into the formulas.
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1) The document discusses the distance and midpoint formulas. The distance formula calculates the distance between two points on a coordinate plane using their x and y coordinates, while the midpoint formula calculates the midpoint between two points.
2) An example uses the distance formula to calculate the distance between points (2, -6) and (-3, 6), finding it to be 13.
3) Another example finds the midpoint between points (1, 2) and (-5, 6) to be (-2, 8) using the midpoint formula.
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1. The document discusses finding the midpoint and distance between points in a coordinate plane. It provides examples and steps for using the midpoint formula and distance formula.
2. The midpoint formula is used to find the midpoint between two points by taking the average of the x-coordinates and y-coordinates. The distance formula and Pythagorean theorem can both be used to find the distance between points.
3. Examples include finding midpoints and distances between points, as well as applications to sports such as finding the distance of a throw between bases in a baseball diamond.
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The system of equations is solved as follows:
1) x + 2y = 5 and y = 3x - 1 are substituted into each other and simplified
2) This results in 7x = 7, so x = 1
3) Substituting x = 1 into y = 3x - 1 gives y = 2
4) Therefore, the solution is (1, 2).
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The document provides information about the standard equation of a circle given its center and tangency condition. It gives the equation (x – 5)2 + (y + 6)2 = 36 for the circle with center (5,-6) tangent to the x-axis. It then explains how to rewrite this in general form as x2 + y2 - 10x +12y+25 =0. It also gives the general form of the equation of a circle Ax2 + By2 + Cx + Dy + E = 0 and examples of identifying the center and radius from equations in general form.
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This document discusses graphing circles and converting between graphing form and standard form equations of circles. It provides examples of writing the equation of a circle given its center and radius in graphing form. It also shows how to find the center, radius, and equation of a circle given the standard form equation. The key points are:
- In graphing form, the center is denoted as (h, k) and the equation is (x - h)2 + (y - k)2 = r2
- To convert to standard form, complete the square and factor the left side, moving any constant to the right side.
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Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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This document provides a study package on circles for a mathematics class. It begins with an index listing the topics covered, which include theory, revision, exercises, assertion and reason questions, and past examination questions. It then covers circle theory, equations of circles in various forms including parametric and Cartesian, intercepts made by circles on axes, the position of points with respect to circles, lines and circles, and tangents to circles. Examples are provided to illustrate each concept. The document is intended to be a comprehensive resource for students to learn about circles.
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This document provides solutions to mathematical problems involving straight lines and circles. Some key details:
- Problem 1 involves solving a system of linear equations to find the value of a variable in the first quadrant.
- Problem 2 describes the family of lines passing through a given point and the equation of the angle bisector.
- Problem 3 discusses how the area between a circle and line is maximized when the line acts as the diameter.
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This document explains the distance and midpoint formulas. The distance formula, derived from the Pythagorean theorem, is used to find the distance between two points with coordinates (x1, y1) and (x2, y2). It is defined as d = √((y2 - y1)2 + (x2 - x1)2). The midpoint formula finds the midpoint between two points; the x-coordinate is (x1 + x2)/2 and the y-coordinate is (y1 + y2)/2. Examples are provided to demonstrate calculating the distance between points and finding the midpoint.
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The document discusses distance and midpoint formulas. It provides the formulas for calculating the distance between two points and finding the midpoint between two points. Examples are included to demonstrate using the formulas to find the distance and midpoint for pairs of points in a coordinate plane. The key formulas are:
Distance Formula: d = √(x2 - x1)2 + (y2 - y1)2
Midpoint Formula: Midpoint = (x1 + x2)/2, (y1 + y2)/2
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This document discusses formulas for finding the distance and midpoint between two points. It provides the distance formula, midpoint formula, and worked examples finding the distance and midpoint between points like (-3, 1) and (2, 3). It also includes a bonus problem asking to find all possible values of a given the distance between (4, a) and (1, 6) is 5 units. An alternate method for finding distance by graphing the points and using the lengths of the sides of the right triangle formed is also described.
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The document presents several problems involving finding equations of circles given properties such as the center and radius, points the circle passes through, tangency to lines, etc. There are over 20 subproblems across 4 sections - A focuses on finding center and radius of circles given equations, B focuses on finding equations of circles given properties like center and radius or points, C involves circles tangent to given lines, and D analyzes if equations represent circles, points or empty sets and finds properties if they are circles.
3. apply distance and midpoint
This document provides formulas and examples for calculating the distance and midpoint between two points.
The distance formula is used to find the distance between two points (x1, y1) and (x2, y2). The midpoint formula finds the midpoint between two points by taking the average of the x-coordinates and y-coordinates.
Several examples demonstrate using the distance and midpoint formulas to find distances, midpoints, and missing endpoint coordinates. The key steps are to work inside parentheses, do exponents before addition, and plug points into the formulas.
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1) The document discusses the distance and midpoint formulas. The distance formula calculates the distance between two points on a coordinate plane using their x and y coordinates, while the midpoint formula calculates the midpoint between two points.
2) An example uses the distance formula to calculate the distance between points (2, -6) and (-3, 6), finding it to be 13.
3) Another example finds the midpoint between points (1, 2) and (-5, 6) to be (-2, 8) using the midpoint formula.
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1. The document discusses finding the midpoint and distance between points in a coordinate plane. It provides examples and steps for using the midpoint formula and distance formula.
2. The midpoint formula is used to find the midpoint between two points by taking the average of the x-coordinates and y-coordinates. The distance formula and Pythagorean theorem can both be used to find the distance between points.
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The system of equations is solved as follows:
1) x + 2y = 5 and y = 3x - 1 are substituted into each other and simplified
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3) Substituting x = 1 into y = 3x - 1 gives y = 2
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This document discusses the metric K-center problem and algorithms for approximating its solution. It shows that metric K-center is NP-hard by reducing it to the dominating set problem. It also proves that metric K-center cannot be approximated within a factor of 2, unless P=NP. The document then describes Gonzalez's algorithm, which achieves an approximation ratio of 2 by greedily selecting centers to maximize the minimum distance from existing centers.
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The document discusses determining the position of a point relative to a circle. It provides the equation for a circle and explains that a point can be inside, outside, or on the curve of the circle. Examples are given to demonstrate how to calculate the position of a point using the circle equation and comparing it to the radius. The document concludes by thanking the audience.
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The document discusses the distance formula and how to calculate the distance between two points. It provides the formula: Distance = √(x2 - x1)2 + (y2 - y1)2. Several examples are shown of using the distance formula to find the distance between points. The document also covers finding the midpoint between two points using the formula: Midpoint = (x1 + x2)/2, (y1 + y2)/2.
Cricle.pptx
Circle, Definition, Equation of circle whose center and radius is known, General equation of a circle, Equation of circle passing through three given points, Equation of circle whose diameters is line joining two points (x1, y1) & (x2,y2), Tangent and Normal to a given circle at given point.
Bressenham’s Midpoint Circle Drawing Algorithm
This document explains Bresenham's midpoint circle drawing algorithm. It defines a circle as all points equidistant from a center point. The algorithm uses integer arithmetic to iteratively determine the next pixel on the circle circumference. It starts at an initial point and calculates a decision parameter P to determine if the next point is above or below along the y-axis. It tabulates an example of drawing a circle of radius 10 centered at the origin to demonstrate how P is used to progress around the circle octant by octant.
Ceramah Add Mth
This document provides guidance on solving additional mathematics problems and preparing for the additional mathematics paper 1 and paper 2 exams. It discusses exam formats, common mistakes students make, key strategies for achieving high marks, and examples of solved exam questions. The document emphasizes understanding the problem, planning a strategy, checking answers, and showing working clearly. It also provides tips on time management and the types of questions that may appear.
3.3 the fundamental theorem of algebra t
The document discusses the fundamental theorem of algebra through examples of factoring polynomials and finding their roots. It provides exercises to factor polynomials into real factors and list all real and complex roots. It also gives exercises to find the real polynomial P(x) given its roots, degree, and an initial condition of P(x) at a specified value of x. The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Circles
This document provides information about circles and their relationships to other geometric concepts:
1. It defines key concepts such as the distance formula, points on a circle, the equation of a circle with the center at the origin or other point (x,y), and the general equation of a circle.
2. It discusses the intersection of circles and lines, including finding the points of intersection using substitution and solving quadratics. The discriminant is used to determine the number of intersection points.
3. Tangents are defined as lines that intersect a circle at only one point and formulas are given for finding the equation of a tangent line to a circle.
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Add Maths Module
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
Presentation on calculus
This document provides information about an upcoming mathematics presentation including:
- A list of 12 group members presenting with their names and student IDs
- 16 topics that will be covered in the presentation ranging from distance between points and planes to surface integrals
- Sample problems and solutions for several topics like finding the distance between a point and plane, parallel planes, a point and line in space, and vector valued functions. Formulas and definitions are provided for concepts like velocity, acceleration, tangent vectors, arc length, and curvature.
mathemathics + Straight line equation
The document discusses key concepts in coordinate geometry, including:
1) How to calculate the distance between two points using their coordinates. The distance formula is given as the square root of the sum of the squared differences between the x- and y-coordinates.
2) How to find the midpoint between two points by taking the average of their x- and y-coordinates. The midpoint formula is given.
3) How to find a point that divides a line segment between two end points in a given ratio of distances, using a formula that involves the x- and y-coordinates and the ratio. Examples of each concept are worked out.
Presentacion unidad 4
This document contains solutions to exercises on conic sections (hyperbolas, ellipses, circles, and parabolas) from a geometry guide.
The solutions include finding the center, vertices, foci, and eccentricity of various hyperbolas and ellipses given in standard form. One example given is a circle, for which the center and radius are identified. Another example is completed by rewriting the equation in canonical form.
The purpose is to understand these geometry topics for future professional careers by solving the guide's problems and verifying answers using GeoGebra.
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Kelompok 1
- 1. MATH TASK GROUP 1 DIAN PERMANA ANNISA INDAH REZA DHIA ATIKAH ALIYYU USWATUN HASANAH WAN ELSA
- 2. The Circle Equation with Center P(a,b) and radius r ! The general equation distance is : (x2-x1)2 + (y2-y1)2 =r2 So in the circle, x1 = a y1 = b x2 = x y2 = y Then the circle equation become : (x-a)2 + (y-b)2= r2 y T(x,y) r P(a,b) x
- 3. Example . . . Determine the circle equation with center P(-2,3) with the following requirement : a. Passing through point Q(4,-1) b. Meet an exatly one point X c. Meet an exatly one point Y
- 4. Answers . . . 1. a. Q(4,-1) P(2,-3) (4-2)2 +(-3+1)2 = r2 (2)2 + (-2)2 = r2 8 = r2 (x-2)2 + (y+1)2 = 8 b. Meet an exatly one point X means r = b = 3 (x-2) + (y+1) = 9 c. Meet an exatly one point Y means r = a = 2 (x-2) + (y+1) = 4