Download - Circles - analysis problems
Circles - Analysis Problems
Mathematics 4
August 15, 2011
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Circle Analysis
Example 1
A 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 theradius and point of tangency.
2. To get the value of the radius, we need to find the coordinates ofthe point of tangency.
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Circle Analysis
Example 1
A 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 theradius and point of tangency.
2. To get the value of the radius, we need to find the coordinates ofthe point of tangency.
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Circle Analysis
Example 1
A 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 theradius and point of tangency.
2. To get the value of the radius, we need to find the coordinates ofthe point of tangency.
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Circle Analysis
Example 1
A 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 theradius and point of tangency.
2. To get the value of the radius, we need to find the coordinates ofthe point of tangency.
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Circle Analysis
Example 1
A 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 theradius and point of tangency.
2. To get the value of the radius, we need to find the coordinates ofthe point of tangency.
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Circle Analysis - Finding the required radius
• Center at (2, 1)
• Tangent to y = x+ 2
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Circle Analysis - Finding the required radius
• Center at (2, 1)
• Tangent to y = x+ 2
• Find the equation of the lineperpendicular to the tangentline and passing through thecenter of the circle.
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Circle Analysis - Finding the required radius
• Center at (2, 1)
• Tangent to y = x+ 2
• Find the equation of the lineperpendicular to the tangentline and passing through thecenter of the circle.
• Find the intersection of thisline with the original line usingsystems of equations to getthe point of tangency.
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Circle Analysis - Finding the required radius
• Center at (2, 1)
• Tangent to y = x+ 2
• Find the equation of the lineperpendicular to the tangentline and passing through thecenter of the circle.
• Find the intersection of thisline with the original line usingsystems of equations to getthe point of tangency.
• Find the distance from P andC to get the radius.
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Circle Analysis - Finding the required radius
(x− 2)2 + (y − 1) = 92
• Center at (2, 1)
• Tangent to y = x+ 2
• Find the equation of the lineperpendicular to the tangentline and passing through thecenter of the circle.
• Find the intersection of thisline with the original line usingsystems of equations to getthe point of tangency.
• Find the distance from P andC to get the radius.
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Recitation Problem
For 2 reci points
Find 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
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Recitation Problem
For 2 reci points
Find 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.
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Circle Analysis
Example 2
Find 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 = r2
2. Three different points satisfying this equation.
What do we need to do? → Find the values for h, k and r2.
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Circle Analysis
Example 2
Find 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 = r2
2. Three different points satisfying this equation.
What do we need to do? → Find the values for h, k and r2.
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Circle Analysis
Example 2
Find 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 = r2
2. Three different points satisfying this equation.
What do we need to do? → Find the values for h, k and r2.
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Circle Analysis
Example 2
Find 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 = r2
2. Three different points satisfying this equation.
What do we need to do? → Find the values for h, k and r2.
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Circle Analysis
Example 2
Find 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 = r2
2. Three different points satisfying this equation.
What do we need to do? → Find the values for h, k and r2.
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Circle Analysis
Example 2
Find 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 = r2
2. Three different points satisfying this equation.
What do we need to do?
→ Find the values for h, k and r2.
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Circle Analysis
Example 2
Find 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 = r2
2. Three different points satisfying this equation.
What do we need to do? → Find the values for h, k and r2.
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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 = 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
2 = 3 (3− h)2 + (5− k)2 = (7− h)2 + (3− k)2
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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 = 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
2 = 3 (3− h)2 + (5− k)2 = (7− h)2 + (3− k)2
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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 = 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
2 = 3 (3− h)2 + (5− k)2 = (7− h)2 + (3− k)2
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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 = 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)
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Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).
Solving Equations A and B simultaneously:
3h+ k = 9
2h− k = 6
We 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 = r2
Final standard equation: (x− 3)2 + y2 = 25
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Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).
Solving Equations A and B simultaneously:
3h+ k = 9
2h− k = 6
We 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 = r2
Final standard equation: (x− 3)2 + y2 = 25
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Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).
Solving Equations A and B simultaneously:
3h+ k = 9
2h− k = 6
We 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 = r2
Final standard equation: (x− 3)2 + y2 = 25
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Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).
Solving Equations A and B simultaneously:
3h+ k = 9
2h− k = 6
We 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 = r2
Final standard equation: (x− 3)2 + y2 = 25
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Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).
Solving Equations A and B simultaneously:
3h+ k = 9
2h− k = 6
We 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 = r2
Final standard equation: (x− 3)2 + y2 = 25
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Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).
Solving Equations A and B simultaneously:
3h+ k = 9
2h− k = 6
We 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 = r2
Final standard equation: (x− 3)2 + y2 = 2513 of 21
Recitation Problem
Reci Problem 2Find the general equation of the circle containing the pointsA(−5, 0), B(1, 0), and C(−2,−3).
x2 + y2 + 4x− 5 = 0
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Recitation Problem
Reci Problem 2Find the general equation of the circle containing the pointsA(−5, 0), B(1, 0), and C(−2,−3).
x2 + y2 + 4x− 5 = 0
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis
Example 3
Find 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 themidpoints, which is the center.
• Find the radius by getting the distance from the center to one ofthe points in the circle.
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Circle Analysis - Finding the required radius
• Circle passes throughA(0, 4), B(3, 5) and C(7, 3)
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Circle Analysis - Finding the required radius
• Circle passes throughA(0, 4), B(3, 5) and C(7, 3)
• Find the midpoints of twochords in this circle.
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Circle Analysis - Finding the required radius
• Circle passes throughA(0, 4), B(3, 5) and C(7, 3)
• Find the midpoints of twochords in this circle.
• Find the equation of theperpendicular bisectors passingthrought the midpoints.
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Circle Analysis - Finding the required radius
• Circle passes throughA(0, 4), B(3, 5) and C(7, 3)
• Find the midpoints of twochords in this circle.
• Find the equation of theperpendicular bisectors passingthrought the midpoints.
• Find the intersection of theperpendicular bisectors.
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Circle Analysis - Finding the required radius
(x− 3)2 + y2 = 25
• Circle passes throughA(0, 4), B(3, 5) and C(7, 3)
• Find the midpoints of twochords in this circle.
• Find the equation of theperpendicular bisectors passingthrought the midpoints.
• Find the intersection of theperpendicular bisectors.
• Find the radius and constructthe circle equation.
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Recitation Problem
Reci 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 = 16
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Recitation Problem
Reci 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 = 16
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