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Circles and Tangent Lines

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  • 1. Circles and Tangent Lines Mathematics 4 August 22, 20111 of 15
  • 2. Distance of a point from a lineThe perpendicular distance of a point (h, k) from a lineAx + By + C = 0 is given by the formula: |Ah + Bk + C| d= √ A2 + B 2 2 of 15
  • 3. Recall first analysis problem on circles:Example 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle. 3 of 15
  • 4. Recall first analysis problem on circles:Example 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle. • Given C(2, 1) and tangent line x − y + 2 = 0. • • 3 of 15
  • 5. Recall first analysis problem on circles:Example 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle. • Given C(2, 1) and tangent line x − y + 2 = 0. • Use the formula for the perpendicular distance: r = |Ah+Bk+C| √ A2 +B 2 • 3 of 15
  • 6. Recall first analysis problem on circles:Example 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle. • Given C(2, 1) and tangent line x − y + 2 = 0. • Use the formula for the perpendicular distance: r = |(1·2)+(−1·1)+2| √2 2 1 +(−1) • 3 of 15
  • 7. Recall first analysis problem on circles:Example 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle. • Given C(2, 1) and tangent line x − y + 2 = 0. • Use the formula for the perpendicular distance: r = |(1·2)+(−1·1)+2| = √ √2 2 3 2 1 +(−1) • 3 of 15
  • 8. Recall first analysis problem on circles:Example 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle. • Given C(2, 1) and tangent line x − y + 2 = 0. • Use the formula for the perpendicular distance: r = |(1·2)+(−1·1)+2| = √ √2 2 3 2 1 +(−1) • (x − 2)2 + (y − 1)2 = 9 2 3 of 15
  • 9. Updated Tournament Rules1. Each section can block 15 other students from the other sections from winning a round.2. Each section can declare 5 students from their section as automatic round winners.3. Blocking trumps automatic win. 4 of 15
  • 10. Quiz 3Find the equations of the circles with the following properties. Showcomplete solutions.1. (HW2 Problem 6) Concentric with the circle x2 + y 2 − 6x + 2y − 15 = 0 and tangent to the line 5x + 12y + 10 = 0.2. Passing through the points (2, 3), (4, 5), and (0, −3). 5 of 15
  • 11. Two points and center on a lineExample 2Find the SE of the circle passingthrough the points (−4, −2) and(2, 0), and whose center lies onthe line y = 2 x − 19 . 5 2Use an algebraic approach. 6 of 15
  • 12. Through (−4, −2) and (2, 0), center at 5x − 2y = 19 Since C is the center, then AC = BC = r.7 of 15
  • 13. Through (−4, −2) and (2, 0), center at 5x − 2y = 19 Since C is the center, then AC = BC = r. Equation 1: 3h + k = −47 of 15
  • 14. Through (−4, −2) and (2, 0), center at 5x − 2y = 19 Since C is the center, then AC = BC = r. Equation 1: 3h + k = −4 Since C is on the line, then h and k satisfy the equation 5x − 2y = 19.7 of 15
  • 15. Through (−4, −2) and (2, 0), center at 5x − 2y = 19 Since C is the center, then AC = BC = r. Equation 1: 3h + k = −4 Since C is on the line, then h and k satisfy the equation 5x − 2y = 19. Equation 2: 5h − 2k = 197 of 15
  • 16. Through (−4, −2) and (2, 0), center at 5x − 2y = 19 Solving the system of equations: Equation 1: 3h + k = −4 Equation 2: 5h − 2k = 198 of 15
  • 17. Through (−4, −2) and (2, 0), center at 5x − 2y = 19 Solving the system of equations: Equation 1: 3h + k = −4 Equation 2: 5h − 2k = 19 Center is at C(1, −7)8 of 15
  • 18. Through (−4, −2) and (2, 0), center at 5x − 2y = 19 Solving the system of equations: Equation 1: 3h + k = −4 Equation 2: 5h − 2k = 19 Center is at C(1, −7) Use distance formula to get r = AC8 of 15
  • 19. Through (−4, −2) and (2, 0), center at 5x − 2y = 19 Solving the system of equations: Equation 1: 3h + k = −4 Equation 2: 5h − 2k = 19 Center is at C(1, −7) Use distance formula to get r = AC Final Equation: (x − 1)2 + (y + 7)2 = 508 of 15
  • 20. Two points and center on a lineHomework 4Show that the SE of the circle passing through the points (−4, −2)and (2, 0), and whose center lies on the line y = 2 x − 19 . is 5 2(x − 1)2 + (y + 7)2 = 50Use a GEOMETRIC approach. Show complete solutions. 9 of 15
  • 21. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2).10 of 15
  • 22. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2). 1. Get the perpendicular line L1 to x + 7y − 2 = 0 passing through A(2, 0).10 of 15
  • 23. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2). 1. Get the perpendicular line L1 to x + 7y − 2 = 0 passing through A(2, 0).10 of 15
  • 24. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2). 1. Get the perpendicular line L1 to x + 7y − 2 = 0 passing through A(2, 0). 2. Get the perpendicular line L2 to AB passing through the midpoint.10 of 15
  • 25. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2). 1. Get the perpendicular line L1 to x + 7y − 2 = 0 passing through A(2, 0). 2. Get the perpendicular line L2 to AB passing through the midpoint.10 of 15
  • 26. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2). 1. Get the perpendicular line L1 to x + 7y − 2 = 0 passing through A(2, 0). 2. Get the perpendicular line L2 to AB passing through the midpoint. 3. Get the intersection of the L1 and L2 to get the center.10 of 15
  • 27. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2). 1. Get the perpendicular line L1 to x + 7y − 2 = 0 passing through A(2, 0). 2. Get the perpendicular line L2 to AB passing through the midpoint. 3. Get the intersection of the L1 and L2 to get the center.10 of 15
  • 28. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2). 1. Get the perpendicular line L1 to x + 7y − 2 = 0 passing through A(2, 0). 2. Get the perpendicular line L2 to AB passing through the midpoint. 3. Get the intersection of the L1 and L2 to get the center. 4. Find the distance from the center to one of the points to get the10 of 15 radius.
  • 29. Homework 5Problem 1Find the standard equation of the circle tangent to the linex + 7y − 2 = 0 at A(2, 0) and passing through B(−4, −2). 1. Get the perpendicular line L1 to x + 7y − 2 = 0 passing through A(2, 0). 2. Get the perpendicular line L2 to AB passing through the midpoint. 3. Get the intersection of the L1 and L2 to get the center. 4. Find the distance from the center to one of the points to get the10 of 15 radius.
  • 30. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0).11 of 15
  • 31. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0). 1. Get the equation of the line L1 passing through D(−5, −1) and A(−4, −2) .11 of 15
  • 32. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0). 1. Get the equation of the line L1 passing through D(−5, −1) and A(−4, −2) .11 of 15
  • 33. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0). 1. Get the equation of the line L1 passing through D(−5, −1) and A(−4, −2) . 2. Get the perpendicular line L2 to AB passing through the midpoint.11 of 15
  • 34. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0). 1. Get the equation of the line L1 passing through D(−5, −1) and A(−4, −2) . 2. Get the perpendicular line L2 to AB passing through the midpoint.11 of 15
  • 35. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0). 1. Get the equation of the line L1 passing through D(−5, −1) and A(−4, −2) . 2. Get the perpendicular line L2 to AB passing through the midpoint. 3. Get the intersection of the L1 and L2 to get the center.11 of 15
  • 36. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0). 1. Get the equation of the line L1 passing through D(−5, −1) and A(−4, −2) . 2. Get the perpendicular line L2 to AB passing through the midpoint. 3. Get the intersection of the L1 and L2 to get the center.11 of 15
  • 37. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0). 1. Get the equation of the line L1 passing through D(−5, −1) and A(−4, −2) . 2. Get the perpendicular line L2 to AB passing through the midpoint. 3. Get the intersection of the L1 and L2 to get the center. 4. Find the distance from the center11 of 15 to one of the points to get the radius.
  • 38. Homework 5Problem 2Find the standard equation of the circle externally tangent to thecircle (x + 5)2 + (y + 1)2 = 2 at A(−4, −2) and passing throughB(2, 0). 1. Get the equation of the line L1 passing through D(−5, −1) and A(−4, −2) . 2. Get the perpendicular line L2 to AB passing through the midpoint. 3. Get the intersection of the L1 and L2 to get the center. 4. Find the distance from the center11 of 15 to one of the points to get the radius.
  • 39. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency.12 of 15
  • 40. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency. 1. Get the equation of the line L3 equidistant from L1 and L2 .12 of 15
  • 41. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency. 1. Get the equation of the line L3 equidistant from L1 and L2 .12 of 15
  • 42. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency. 1. Get the equation of the line L3 equidistant from L1 and L2 . 2. Get the perpendicular line L4 to L1 passing through A(2, 1).12 of 15
  • 43. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency. 1. Get the equation of the line L3 equidistant from L1 and L2 . 2. Get the perpendicular line L4 to L1 passing through A(2, 1).12 of 15
  • 44. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency. 1. Get the equation of the line L3 equidistant from L1 and L2 . 2. Get the perpendicular line L4 to L1 passing through A(2, 1). 3. Get the intersection of the L3 and L4 to get the center.12 of 15
  • 45. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency. 1. Get the equation of the line L3 equidistant from L1 and L2 . 2. Get the perpendicular line L4 to L1 passing through A(2, 1). 3. Get the intersection of the L3 and L4 to get the center.12 of 15
  • 46. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency. 1. Get the equation of the line L3 equidistant from L1 and L2 . 2. Get the perpendicular line L4 to L1 passing through A(2, 1). 3. Get the intersection of the L3 and L4 to get the center. 4. Find the distance from the center A(2, 1) to get the radius.12 of 15
  • 47. Homework 5Problem 3Find the standard equation of the circle tangent to the linesL1 : 2x + y − 5 = 0 and L2 : 2x + y + 15 = 0 if A(2, 1) is one pointof tangency. 1. Get the equation of the line L3 equidistant from L1 and L2 . 2. Get the perpendicular line L4 to L1 passing through A(2, 1). 3. Get the intersection of the L3 and L4 to get the center. 4. Find the distance from the center A(2, 1) to get the radius.12 of 15
  • 48. Example 3Quadratic SystemsFind the standard equation of the circle passing through the point(7, 9) tangent to the x-axis and has its center on the linex−y+1=013 of 15
  • 49. Example 3Quadratic SystemsFind the standard equation of the circle passing through the point(7, 9) tangent to the x-axis and has its center on the linex−y+1=014 of 15
  • 50. Example 4Quadratic SystemsFind the standard equation of the circle tangent to the line √x − 2y = 3 at A(−1, −2) and having a radius 515 of 15