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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

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LC.01.3 - The Circle

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  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

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