Circles
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This document defines circles geometrically as the result of a cone intersecting with a plane, and algebraically as the set of points equidistant from a fixed center point. It provides the standard equation for a circle given the center (h,k) and radius r: (x-h)2 + (y-k)2 = r2. Examples are given of writing the equation of a circle and finding its center and radius given parts of the equation. The process for finding the equation of a circle given its diameter is also described.
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G10 Math Q2- Week 8- Equation of a Circle.pptx
This document discusses the standard and general forms of equations for circles. It provides examples of writing the standard and general form equations given the center and radius of various circles. Key points 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.
- The general form is Ax2 + By2 + Cx + Dy + E = 0, where the coefficients can be used to find the center and radius.
- Examples are worked through of writing circle equations in both standard and general form given attributes of the circle like the center and radius.
Lesson 14: Equation of a Circle
The document discusses finding equations of circles given information about their properties. It provides examples of finding the equation of a circle given its center and radius, center and a point on the circle, or the endpoints of its diameter. The key steps are to use the standard form equation x-h^2 + y-k^2 = r^2, where (h,k) is the center and r is the radius, and substitute the given values to find the equation.
Geometry unit 12.5
The document discusses circles in the coordinate plane. It provides examples of writing the equation of a circle given its center and radius or given two points it passes through. It also gives examples of graphing circles given their equations by identifying the center and radius. Additionally, it presents an example of using three points and their perpendicular bisectors to find the center of a circle passing through those three points, and applies this to solving a problem about finding the location of a structure.
48 circle part 1 of 2
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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Circles
- 2. CIRCLES • Geometric: the result of a cone and a plane intersecting each other • Algebraic: set of points on a plane that are equidistant from a fixed point on the same plane (center/origin)
- 3. x y Radius (r) Center (h,k) - The distance from the center of the circle (h,k) to any point on the circle (x,y). RADIUS (x,y)
- 4. Equation of the Circle • The center-radius or standard form: (𝑥 − ℎ)2+(𝑦 − 𝑘)2 = 𝑟2 where (h,k) is the center and r is the radius • Derived from the Pythagorean Theorem 𝑎2 + 𝑏2 = 𝑟2 𝑎 𝑏
- 5. Find the center and the radius for each of the following circles. 1. (𝑥 − 4)2 +(𝑦 − 6)2 = 25 2. (𝑥 − 3)2 +(𝑦 + 1)2 = 16 3. (𝑥 + 5)2 + 𝑦 − 2 = 15 4. 𝑥2 + (𝑦 + 7)2 = 18 ans. center is (4,6) & r = 5 ans. center is (3,-1) & r = 4 ans. center is (-5,2) & r = 15 ans. center is (0,-7) & r = 3 2
- 6. Write the equation of the circle given the center and the radius. 1. C (2,-7) & r = 3 (𝑥 − 2)2+(𝑦 + 7)2= 9 2. C (-3, -3) & r = 10 (𝑥 + 3)2+(𝑥 + 3)2= 10
- 7. Write the equation of a circle whose diameter is the line segment joining A(-3,-4) and B(4,3). • What must you find first? The center and the radius. • How can you find the center? The center is the midpoint of the segment. (½ , - ½ ) • How can you find the radius? The radius is the distance from the center to a point on the circle. Use the distance formula. r = 7 1 2 2 2 1 1 49 2 2 2 x y
- 8. Find the equation of the circle. 𝑥2 + 𝑦2 + 6𝑥 − 4𝑦 − 12 = 0 Hint: You must complete the square 2 2 6 ___ 4 ___ 12x x y y 2 2 6 9 4 4 12 9 4x x y y 2 2 ( 3) ( 2) 25x y