Circles and Tangent Lines

Circles and Tangent Lines

Mathematics 4

August 22, 2011

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Distance of a point from a line

The perpendicular distance of a point (h, k) from a lineAx+By + C = 0 is given by the formula:

d =|Ah+Bk + C|√

A2 +B2

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Recall first analysis problem on circles:

Example 1

A circle with center (2, 1) is tangent to the line y = x+ 2. Find theequation of this circle.

••

•

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Recall first analysis problem on circles:

Example 1

A circle with center (2, 1) is tangent to the line y = x+ 2. Find theequation of this circle.

• Given C(2, 1) and tangent linex− y + 2 = 0.

••

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Recall first analysis problem on circles:

Example 1

A circle with center (2, 1) is tangent to the line y = x+ 2. Find theequation of this circle.

• Given C(2, 1) and tangent linex− y + 2 = 0.

• Use the formula for theperpendicular distance:r = |Ah+Bk+C|√

A2+B2

•

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Recall first analysis problem on circles:

Example 1

A circle with center (2, 1) is tangent to the line y = x+ 2. Find theequation of this circle.

• Given C(2, 1) and tangent linex− y + 2 = 0.

• Use the formula for theperpendicular distance:r = |(1·2)+(−1·1)+2|√

12+(−1)2

•

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Recall first analysis problem on circles:

Example 1

A circle with center (2, 1) is tangent to the line y = x+ 2. Find theequation of this circle.

• Given C(2, 1) and tangent linex− y + 2 = 0.

• Use the formula for theperpendicular distance:r = |(1·2)+(−1·1)+2|√

12+(−1)2= 3√

2

•

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Recall first analysis problem on circles:

Example 1

A circle with center (2, 1) is tangent to the line y = x+ 2. Find theequation of this circle.

• Given C(2, 1) and tangent linex− y + 2 = 0.

• Use the formula for theperpendicular distance:r = |(1·2)+(−1·1)+2|√

12+(−1)2= 3√

2

• (x− 2)2 + (y − 1)2 = 92

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Updated Tournament Rules

1. Each section can block 15 other students from the other sectionsfrom winning a round.

2. Each section can declare 5 students from their section asautomatic round winners.

3. Blocking trumps automatic win.

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Quiz 3

Find the equations of the circles with the following properties. Showcomplete solutions.

1. (HW2 Problem 6) Concentric with the circlex2 + y2 − 6x+ 2y − 15 = 0 and tangent to the line5x+ 12y + 10 = 0.

2. Passing through the points (2, 3), (4, 5), and (0,−3).

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Two points and center on a line

Example 2

Find the SE of the circle passingthrough the points (−4,−2) and(2, 0), and whose center lies onthe line y = 5

2x−192 .

Use an algebraic approach.

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Through (−4,−2) and (2, 0), center at 5x− 2y = 19

Since C is the center, thenAC = BC = r.

Equation 1: 3h+ k = −4

Since C is on the line, then h andk satisfy the equation5x− 2y = 19.

Equation 2: 5h− 2k = 19

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Through (−4,−2) and (2, 0), center at 5x− 2y = 19

Since C is the center, thenAC = BC = r.

Equation 1: 3h+ k = −4

Since C is on the line, then h andk satisfy the equation5x− 2y = 19.

Equation 2: 5h− 2k = 19

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Through (−4,−2) and (2, 0), center at 5x− 2y = 19

Since C is the center, thenAC = BC = r.

Equation 1: 3h+ k = −4

Since C is on the line, then h andk satisfy the equation5x− 2y = 19.

Equation 2: 5h− 2k = 19

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Through (−4,−2) and (2, 0), center at 5x− 2y = 19

Since C is the center, thenAC = BC = r.

Equation 1: 3h+ k = −4

Since C is on the line, then h andk satisfy the equation5x− 2y = 19.

Equation 2: 5h− 2k = 19

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Through (−4,−2) and (2, 0), center at 5x− 2y = 19

Solving the system of equations:

Equation 1: 3h+ k = −4Equation 2: 5h− 2k = 19

Center is at C(1,−7)

Use distance formula to getr = AC

Final Equation:(x− 1)2 + (y + 7)2 = 50

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Through (−4,−2) and (2, 0), center at 5x− 2y = 19

Solving the system of equations:

Equation 1: 3h+ k = −4Equation 2: 5h− 2k = 19

Center is at C(1,−7)

Use distance formula to getr = AC

Final Equation:(x− 1)2 + (y + 7)2 = 50

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Through (−4,−2) and (2, 0), center at 5x− 2y = 19

Solving the system of equations:

Equation 1: 3h+ k = −4Equation 2: 5h− 2k = 19

Center is at C(1,−7)

Use distance formula to getr = AC

Final Equation:(x− 1)2 + (y + 7)2 = 50

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Through (−4,−2) and (2, 0), center at 5x− 2y = 19

Solving the system of equations:

Equation 1: 3h+ k = −4Equation 2: 5h− 2k = 19

Center is at C(1,−7)

Use distance formula to getr = AC

Final Equation:(x− 1)2 + (y + 7)2 = 50

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Two points and center on a line

Homework 4Show that the SE of the circle passing through the points (−4,−2)and (2, 0), and whose center lies on the line y = 5

2x−192 . is

(x− 1)2 + (y + 7)2 = 50

Use a GEOMETRIC approach. Show complete solutions.

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Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.10 of 15

Homework 5

Problem 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 tox+ 7y − 2 = 0 passing throughA(2, 0).

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.10 of 15

Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.

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Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.11 of 15

Homework 5

Problem 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) andA(−4,−2) .

2. Get the perpendicular line L2 toAB passing through themidpoint.

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 the

radius.11 of 15

Homework 5

Problem 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 toL1 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.

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Homework 5

Problem 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 toL1 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.

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Homework 5

Problem 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 toL1 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.

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Homework 5

Problem 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 toL1 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.

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Homework 5

Problem 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 toL1 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.

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Homework 5

Problem 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 toL1 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.

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Homework 5

Problem 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 toL1 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.

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Homework 5

Problem 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 toL1 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.

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Homework 5

Problem 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 toL1 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.

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Example 3

Quadratic Systems

Find 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 = 0

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Example 3

Quadratic Systems

Find 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 = 0

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Example 4

Quadratic Systems

Find the standard equation of the circle tangent to the linex− 2y = 3 at A(−1,−2) and having a radius

√5

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