10.7 writing and graphing circles
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The document discusses the standard form of the equation of a circle and how to find the center and radius from the equation. It provides examples of writing the equation given the center and radius, as well as finding the center and radius from an equation. Key aspects covered include: - The standard form of a circle equation is (x - h)2 + (y - k)2 = r2, where (h, k) is the center and r is the radius - You can find the center and radius by comparing the equation to the standard form - Examples are worked through of writing equations given center and radius values and finding the center and radius from equations
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The document discusses the standard form of a circle equation and how to find the center and radius from the equation. It provides examples of writing the equation of a circle given the center and/or radius or a point on the circle. The standard form is (x - h)2 + (y - k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. If the equation is not in standard form, it may need to be completed into a perfect square to extract the center and radius.
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This document discusses the equations of circles, including the equation of a circle with its center at the origin and the equation of a circle with its center located at a point (h,k). It defines a circle as the set of all points equidistant from a fixed center point, with the distance to the center being the radius. The document provides information to graph circles based on their equations.
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Here are the steps to solve these problems:
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(a) The gradient of the diameter AB is (y2-y1)/(x2-x1).
(b) The equation of AB is y-y1 = (y2-y1)/(x2-x1)(x-x1)
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The document discusses the standard form of a circle equation and how to find the center and radius from the equation. It provides examples of writing the equation of a circle given the center and/or radius or a point on the circle. The standard form is (x - h)2 + (y - k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. If the equation is not in standard form, it may need to be completed into a perfect square to extract the center and radius.
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Conic sections are shapes formed by the intersection of a plane and a double cone. A hyperbola occurs if the plane cuts through both cones. A parabola occurs if the plane is parallel to the edge of the cone. An ellipse occurs if the plane is not parallel or cutting through both cones. A circle is a special case of an ellipse where the plane is perpendicular to the altitude of the cone.
mathemaics 10 lesson about cicles. its part
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Circle : Basic concept
This document defines key terms related to circles:
- A circle consists of all points equidistant from a fixed center point.
- The radius is the fixed distance from the center point to any point on the circle.
- A chord connects any two points on a circle. A diameter passes through the center and divides the circle into two equal halves.
- The standard equation of a circle with center (h, k) and radius r is (x - h)2 + (y - k)2 = r2.
Section 1.3 -- The Coordinate Plane
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
2.2 Circles
Write the equation of a circle given the center and radius
Identify the center and radius of a circle in both center-radius and general form
Write the equation of a circle given the center and a point on the circle
1.5 - equations of circles.ppt
This document provides instruction on writing equations of circles. It begins by defining a circle as all points in the xy-plane that are a fixed distance r from a central point (h,k), known as the radius. The standard form of a circle equation is presented as (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. Examples are given of writing circle equations in standard form given the center and radius. Converting a circle equation from general to standard form is also demonstrated through completing the square. Homework problems are assigned from the text.
Math 10_Q21.5 - equations of circles.ppt
This document provides instruction on writing equations of circles. It begins by defining a circle as points that are a fixed distance (the radius) from a central point (the center). It then presents the standard form of a circle equation as (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the center and r is the radius. Several examples are worked through of writing circle equations in standard and general form given the center and radius. The document also explains how to convert a circle equation from general to standard form through completing the square. Homework problems are assigned.
Circle
The document provides a history and overview of circles. It discusses:
1) Euclid defined circles in 300 BC in his work "The Elements", establishing circles as fundamental objects in geometry. He defined a point and line and established early theorems about circles.
2) Key parts of a circle include the radius, diameter, circumference, chord, secant, tangent, arc, sector, and segment. The circumference is related to the diameter by pi.
3) Circles can be defined analytically using equations in the standard "center-radius" form of (x-h)2 + (y-k)2 = r2, where (h,k) are the coordinates of the
Conic Section: Circles (Pre-Calculus).pdf
The document discusses circles and their applications. It begins by defining a circle and describing how circles are formed. It then provides examples of how circles are used in architecture, transportation, photography, and other fields. The remainder of the document discusses finding the equation of a circle, including the standard and general forms of the circle equation. It provides examples of finding the equation given the center and radius and finding the center and radius given the equation. It also demonstrates how to graph circles using software.
Similar to 10.7 writing and graphing circles (20)
10.7 writing and graphing circles
- 1. CIRCLES
- 2. The standard form of the equation of a circle with its center at the origin is Notice that both the x and y terms are squared. Linear equations don’t have either the x or y terms squared. Parabolas have only the x term was squared (or only the y term, but NOT both). r is the radius of the circle so if we take the square root of the right hand side, we'll know how big the radius is.
- 3. Let's look at the equation The center of the circle is at the origin and the radius is 3. Let's graph this circle. This is r 2 so r = 3 Center at (0, 0) Count out 3 in all directions since that is the radius 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
- 4. If the center of the circle is NOT at the origin then the equation for the standard form of a circle looks like this: The center of the circle is at ( h , k ). The center of the circle is at ( h , k ) which is (3,1). Find the center and radius and graph this circle. The radius is 4 This is r 2 so r = 4 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
- 5. If you take the equation of a circle in standard form for example: You can find the center and radius easily. The center is at (-2, 4) and the radius is 2. Remember center is at ( h , k ) with ( x - h ) and ( y - k ) since the x is plus something and not minus, ( x + 2) can be written as ( x - (-2)) This is r 2 so r = 2 (x - (-2))
- 6. Now let's work some examples: Find an equation of the circle with center at (0, 0) and radius 7. Let's sub in center and radius values in the standard form 0 0 7
- 7. Find an equation of the circle with center at (0, 0) that passes through the point (-1, -4). The point (-1, -4) is on the circle so should work when we plug it in the equation: Since the center is at (0, 0) we'll have Subbing this in for r 2 we have:
- 8. Find an equation of the circle with center at (-2, 5) and radius 6 Subbing in the values in standard form we have: -2 5 6
- 9. Find an equation of the circle with center at (8, 2) and passes through the point (8, 0). Subbing in the center values in standard form we have: 8 2 Since it passes through the point (8, 0) we can plug this point in for x and y to find r 2 .
- 10. Identify the center and radius and sketch the graph: To get in standard form we don't want coefficients on the squared terms so let's divide everything by 9. So the center is at (0, 0) and the radius is 8/3. 9 9 9 Remember to square root this to get the radius. 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
- 11. Identify the center and radius and sketch the graph: Remember the center values end up being the opposite sign of what is with the x and y and the right hand side is the radius squared. So the center is at (-4, 3) and the radius is 5. 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8
- 12. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au