1.
Circles - Analysis Problems Mathematics 4 August 15, 20111 of 21
2.
Circle AnalysisExample 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle. 2 of 21
3.
Circle AnalysisExample 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle.What do we need to solve for? 2 of 21
4.
Circle AnalysisExample 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle.What do we need to solve for? → the radius of the circle. 2 of 21
5.
Circle AnalysisExample 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle.What do we need to solve for? → the radius of the circle.What do we know? 2 of 21
6.
Circle AnalysisExample 1A circle with center (2, 1) is tangent to the line y = x + 2. Find theequation of this circle.What do we need to solve for? → the radius of the circle.What do we know?1. The tangent line is perpendicular to the line passing through the radius and point of tangency.2. To get the value of the radius, we need to ﬁnd the coordinates of the point of tangency. 2 of 21
7.
Circle Analysis - Finding the required radius • Center at (2, 1) • Tangent to y = x + 23 of 21
8.
Circle Analysis - Finding the required radius • Center at (2, 1) • Tangent to y = x + 2 • Find the equation of the line perpendicular to the tangent line and passing through the center of the circle.4 of 21
9.
Circle Analysis - Finding the required radius • Center at (2, 1) • Tangent to y = x + 2 • Find the equation of the line perpendicular to the tangent line and passing through the center of the circle. • Find the intersection of this line with the original line using systems of equations to get the point of tangency.5 of 21
10.
Circle Analysis - Finding the required radius • Center at (2, 1) • Tangent to y = x + 2 • Find the equation of the line perpendicular to the tangent line and passing through the center of the circle. • Find the intersection of this line with the original line using systems of equations to get the point of tangency. • Find the distance from P and C to get the radius.6 of 21
11.
Circle Analysis - Finding the required radius • Center at (2, 1) • Tangent to y = x + 2 • Find the equation of the line perpendicular to the tangent line and passing through the center of the circle. • Find the intersection of this line with the original line using systems of equations to get the point of tangency. • Find the distance from P and 9 (x − 2)2 + (y − 1) = 2 C to get the radius.7 of 21
12.
Recitation ProblemFor 2 reci pointsFind the standard equation of a circle tangent to y = 2x + 11 andwhose center is at C(1, 3).• 1 reci point for the point of tangency• 1 reci point for the standard equation 8 of 21
13.
Recitation ProblemFor 2 reci pointsFind the standard equation of a circle tangent to y = 2x + 11 andwhose center is at C(1, 3).• 1 reci point for the point of tangency → P (−3, 5)• 1 reci point for the standard equation → (x − 1)2 + (y − 3)2 = 20. 9 of 21
14.
Circle AnalysisExample 2Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use an algebraic approach.10 of 21
15.
Circle AnalysisExample 2Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use an algebraic approach.What do we need to solve for?10 of 21
16.
Circle AnalysisExample 2Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use an algebraic approach.What do we need to solve for? → the radius and center of the circle.10 of 21
17.
Circle AnalysisExample 2Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use an algebraic approach.What do we need to solve for? → the radius and center of the circle.What do we know?10 of 21
18.
Circle AnalysisExample 2Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use an algebraic approach.What do we need to solve for? → the radius and center of the circle.What do we know?1. The standard equation of the circle is (x − h)2 + (y − k)2 = r22. Three diﬀerent points satisfying this equation.10 of 21
19.
Circle AnalysisExample 2Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use an algebraic approach.What do we need to solve for? → the radius and center of the circle.What do we know?1. The standard equation of the circle is (x − h)2 + (y − k)2 = r22. Three diﬀerent points satisfying this equation.What do we need to do?10 of 21
20.
Circle AnalysisExample 2Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use an algebraic approach.What do we need to solve for? → the radius and center of the circle.What do we know?1. The standard equation of the circle is (x − h)2 + (y − k)2 = r22. Three diﬀerent points satisfying this equation.What do we need to do? → Find the values for h, k and r2 .10 of 21
21.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).Construct 3 equations using the standard equation and each of thethree points.11 of 21
22.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).Construct 3 equations using the standard equation and each of thethree points.1. (0 − h)2 + (4 − k)2 = r22. (3 − h)2 + (5 − k)2 = r23. (7 − h)2 + (3 − k)2 = r211 of 21
23.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3). Construct 3 equations using the standard equation and each of the three points. 1. (0 − h)2 + (4 − k)2 = r2 2. (3 − h)2 + (5 − k)2 = r2 3. (7 − h)2 + (3 − k)2 = r2 Equate the equations since they are all equal to r2 .1 = 2 (0 − h)2 + (4 − k)2 = (3 − h)2 + (5 − k)22 = 3 (3 − h)2 + (5 − k)2 = (7 − h)2 + (3 − k)2 11 of 21
24.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3). Construct 3 equations using the standard equation and each of the three points. 1. (0 − h)2 + (4 − k)2 = r2 2. (3 − h)2 + (5 − k)2 = r2 3. (7 − h)2 + (3 − k)2 = r2 Equate the equations since they are all equal to r2 .1 = 2 (0 − h)2 + (4 − k)2 = (3 − h)2 + (5 − k)2 → 3h + k = 9 (A)2 = 3 (3 − h)2 + (5 − k)2 = (7 − h)2 + (3 − k)2 → 2h − k = 6 (B) 12 of 21
25.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).Solving Equations A and B simultaneously:13 of 21
26.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).Solving Equations A and B simultaneously: 3h + k = 9 2h − k = 613 of 21
27.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).Solving Equations A and B simultaneously: 3h + k = 9 2h − k = 6We get the center to be (3, 0).13 of 21
28.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).Solving Equations A and B simultaneously: 3h + k = 9 2h − k = 6We get the center to be (3, 0).Find the radius by substituting (3, 0) to any of the ﬁrst threeequations we generated.13 of 21
29.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).Solving Equations A and B simultaneously: 3h + k = 9 2h − k = 6We get the center to be (3, 0).Find the radius by substituting (3, 0) to any of the ﬁrst threeequations we generated. (0 − 3)2 + (4 − 0)2 = r2 9 + 16 = r213 of 21
30.
Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).Solving Equations A and B simultaneously: 3h + k = 9 2h − k = 6We get the center to be (3, 0).Find the radius by substituting (3, 0) to any of the ﬁrst threeequations we generated. (0 − 3)2 + (4 − 0)2 = r2 9 + 16 = r2Final standard equation: (x − 3)2 + y 2 = 2513 of 21
31.
Recitation ProblemReci Problem 2Find the general equation of the circle containing the pointsA(−5, 0), B(1, 0), and C(−2, −3).14 of 21
32.
Recitation ProblemReci Problem 2Find the general equation of the circle containing the pointsA(−5, 0), B(1, 0), and C(−2, −3). x2 + y 2 + 4x − 5 = 014 of 21
33.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.15 of 21
34.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.What do we need to solve for?15 of 21
35.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.What do we need to solve for? → the radius and center of the circle.15 of 21
36.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.What do we need to solve for? → the radius and center of the circle.What do we know?15 of 21
37.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.What do we need to solve for? → the radius and center of the circle.What do we know? → The perpendicular bisectors of chords intersectat the center.15 of 21
38.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.What do we need to solve for? → the radius and center of the circle.What do we know? → The perpendicular bisectors of chords intersectat the center.What do we need to do?15 of 21
39.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.What do we need to solve for? → the radius and center of the circle.What do we know? → The perpendicular bisectors of chords intersectat the center.What do we need to do?• Find the equation of the perpendicular bisectors of the midpoints.15 of 21
40.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.What do we need to solve for? → the radius and center of the circle.What do we know? → The perpendicular bisectors of chords intersectat the center.What do we need to do?• Find the equation of the perpendicular bisectors of the midpoints.• Find the intersection of the perpendicular bisectors of the midpoints, which is the center.15 of 21
41.
Circle AnalysisExample 3Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.What do we need to solve for? → the radius and center of the circle.What do we know? → The perpendicular bisectors of chords intersectat the center.What do we need to do?• Find the equation of the perpendicular bisectors of the midpoints.• Find the intersection of the perpendicular bisectors of the midpoints, which is the center.• Find the radius by getting the distance from the center to one of the points in the circle.15 of 21
42.
Circle Analysis - Finding the required radius • Circle passes through A(0, 4), B(3, 5) and C(7, 3)16 of 21
43.
Circle Analysis - Finding the required radius • Circle passes through A(0, 4), B(3, 5) and C(7, 3) • Find the midpoints of two chords in this circle.17 of 21
44.
Circle Analysis - Finding the required radius • Circle passes through A(0, 4), B(3, 5) and C(7, 3) • Find the midpoints of two chords in this circle. • Find the equation of the perpendicular bisectors passing throught the midpoints.18 of 21
45.
Circle Analysis - Finding the required radius • Circle passes through A(0, 4), B(3, 5) and C(7, 3) • Find the midpoints of two chords in this circle. • Find the equation of the perpendicular bisectors passing throught the midpoints. • Find the intersection of the perpendicular bisectors.19 of 21
46.
Circle Analysis - Finding the required radius • Circle passes through A(0, 4), B(3, 5) and C(7, 3) • Find the midpoints of two chords in this circle. • Find the equation of the perpendicular bisectors passing throught the midpoints. • Find the intersection of the perpendicular bisectors. • Find the radius and construct (x − 3)2 + y2 = 25 the circle equation.20 of 21
47.
Recitation ProblemReci Problem 3Find the standard equation of the circle containing the pointsA(2, 8), B(6, 4) and C(2, 0). Use an geometric approach.21 of 21
48.
Recitation ProblemReci Problem 3Find the standard equation of the circle containing the pointsA(2, 8), B(6, 4) and C(2, 0). Use an geometric approach. (x − 2)2 + (y − 4)2 = 1621 of 21