1 / 12

LC.01.3 - The Circle

LC.01.3 - The Circle. MCR3U - Santowski. (A) Review. Locus Definition for a circle : the set of all points that are a constant distance from a fixed point The constant distance is called the radius The fixed point is called the center Length formula is d 2 = x 2 + y 2

hye
Télécharger la présentation

LC.01.3 - The Circle

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. LC.01.3 - The Circle MCR3U - Santowski

  2. (A) Review • Locus Definition for a circle: the set of all points that are a constant distance from a fixed point • The constant distance is called the radius • The fixed point is called the center • Length formula is d2 = x2+ y2 • Completing the Square technique: x2 + 4x - 1  (x2 + 4x + 4 – 4) – 1  (x + 2)2 - 5

  3. (B) Developing Equations for Circles • ex 1. Use the distance formula to develop an equation for the set of points that are 4 units from (0,0) • Length formula is d2 = x2+ y2 • So 42 = (x – 0)2 + (y – 0)2 • So 16 = x2 + y2 is our equation • ex 2. Use the distance formula to develop an equation for the set of points that are 4 units from (2,3) • Length formula is d2 = x2+ y2 • So 42 = (x – 2)2 + (y – 3)2 • So 16 = x2 – 4x + 4 + y2 - 6y + 9 •  So 0 = x2 + y2 – 4x - 6y – 3 is our equation

  4. (B) Developing Equations for Circles

  5. (B) Developing Equations for Circles • ex 3. Use the distance formula to develop an equation for the set of points that are equidistant from (0,0) • Length formula is d2 = x2+ y2 • So r2 = (x – 0)2 + (y – 0)2 • So r2 = x2 + y2 is our equation • ex 4. Use the distance formula to develop an equation for the set of points that are equidistant from (h,k) • Length formula is d2 = x2+ y2 • So r2 = (x – h)2 + (y – k)2

  6. (C) Forms of Equations for Circles • If the center of the circle is at (0,0), the equation of the circle is x2 + y2 = r2 • If the center of the circle is at (h,k), then the equation of the circle is (x – h)2 + (y – k)2= r2 which is referred to as the standard form • An equation in standard form can be expanded and rewritten as x2 + y2 - 2hx - 2ky + C = 0 which is referred to as the general form

  7. (D) Examples • ex 1. Find the equation of the set of points that are 8 units from (2,-3) • ex 2. Given the circle x2 + y2 = 9, determine the new equation in both standard and general form if the original circle is translated 5 units right and 2 units up. • ex 3. Find the center and radius of the circle x2 + y2 - 10x + 4y + 17 = 0 • So complete the square on this equation to change it from general form to standard form • (x2 – 10x + 25 – 25) + (y2 + 4y + 4 – 4) = -17 • (x – 5)2 + (y + 2)2 = -17 + 25 + 4 • (x – 5)2 + (y + 2)2 = 12 • So the center is at (5,-2) and the radius is 12

  8. (E) Examples • ex 4. Find the equation of the circle tangent to the x-axis and with a center at (3,6) • Being tangent to the x-axis means that the x-axis contacts the circle at only one point, and that point would be perpendicular to the x-axis  the line x = 3 (see diagram) • Now find the radius (of 6 units)

  9. (D) Examples • ex 5. Find the equation of the circle that goes through the points A(0,0), B(4,-8), and C(4,0) • We need to introduce some geometric concepts here •  a chord is a line connecting any 2 points in a circle •  the perpendicular bisector of a chord goes through the center of a circle (see diagram on next page  center is at (2,-4) and radius is (22 + 42) • So the equation is (x – 2)2 + (y + 4)2 = 20

  10. (D) Examples

  11. (E) Internet Links • http://analyzemath.com/CircleEq/FindEquationCircle.html from AnalyzeMath • http://www.webmath.com/gcircle.html from Webmath • http://home.alltel.net/okrebs/page61.html from OJK’s Precalculus Math Page

  12. (F) Homework • page 581, Q1-7eol, 10,13,15 from Nelson text

More Related