of 50 /50
Circles and Tangent Lines Mathematics 4 August 22, 2011 1 of 15

# Circles and Tangent Lines

Embed Size (px)

DESCRIPTION

Class lecture

### Text of Circles and Tangent Lines Circles and Tangent Lines

Mathematics 4

August 22, 2011

1 of 15 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

2 of 15 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.

••

3 of 15 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.

••

3 of 15 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

3 of 15 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 of 15 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

3 of 15 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

3 of 15 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.

4 of 15 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).

5 of 15 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.

6 of 15 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

7 of 15 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

7 of 15 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

7 of 15 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

7 of 15 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

8 of 15 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

8 of 15 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

8 of 15 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

8 of 15 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.

9 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

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

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

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

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

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

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

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 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 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

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

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

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

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

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

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

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 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 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.

12 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.

12 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.

12 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.

12 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.

12 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.

12 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.

12 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.

12 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.

12 of 15 Example 3

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

13 of 15 Example 3

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

14 of 15 Example 4

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

√5

15 of 15 ##### Geometry 9.5 Tangents to Circles. Objectives Identify segments and lines related to circles. Use properties of a tangent to a circle
Documents ##### CONSTRUCTING TANGENT LINES - Classroom Blogplanemath.weebly.com/uploads/1/3/5/1/13515714/... · 3.3.1: Constructing Tangent Lines 3 Constructing a Tangent at a Point on a Circle Using
Documents ##### Circles Chapter 10 Essential Questions How do I identify segments and lines related to circles? How do I use properties of a tangent to a circle?
Documents ##### Tangent Lines - GeometryCoach.com€¦ · 12-04-2017  · Tangent Lines Unit 12 Lesson 1 . TANGENT LINES Students will be able to: Understand the theorem of tangent lines in a circle,
Documents ##### CIRCLES AND VOLUME Lesson 3: Constructing Tangent Linesfortclass.weebly.com/uploads/1/3/1/2/13127295/unit_3-3.pdf6/*5 t CIRCLES AND VOLUME Lesson 3: Constructing Tangent Lines Common
Documents ##### Properties of Circles - Mrs. Luthi's geometrymrsluthismath.weebly.com/.../2/23420766/chpt_10_properties_of_circles.pdf · 650 Chapter 10 Properties of Circles 10.1 Explore Tangent
Documents ##### From Math V3 Tangent Lines From Math 2220 Class 7pi.math.cornell.edu/~back/m222_f14/slides/sep12_v3.pdf · 2015. 1. 15. · From Math 2220 Class 7 V3 Tangent Lines Chain Rule Tangent
Documents ##### 10.1 Tangents to Circles. Objectives/Assignment Students will learn to Identify segments and lines related to circles. Use properties of a tangent to
Documents ##### Circles NAME 12.1 Lines that Intersect Circles (1) Objectives: … · 2018-04-05 · GEO: Unit 8 – Circles NAME _____ 12.1 – Lines that Intersect Circles (1) Objectives: Identify
Documents ##### CIRCLES - CIHS Geometrycihsgeometry.weebly.com/uploads/1/7/5/7/17574061/chapter_10_no... · (tangent circles) 1 1 x A y B ... The point at which a tangent line intersects the circle
Documents ##### Circles - White Plains Middle School€¦ · SWBAT: Apply the Power Theorems 1. Given 2 ... The models below show segment relationships in circles. Secant-Tangent Secant ÂX and tangent
Documents ##### Exit Ticket Packet - Amazon Web Services · Lesson 19: Equations for Tangent Lines to Circles Exit Ticket Consider the circle (𝑥𝑥+ 2)2+ (𝑦𝑦−3)2= 9. There are two lines
Documents ##### The Slope of a Curve or How Secant Lines become Tangent Lines
Documents ##### I. Horizontal and Vertical Tangent Lines How to find them ... · PDF fileI. Horizontal and Vertical Tangent Lines ... Tangent line vertical when dy dx ... A conical tank is leaking
Documents ##### Tangent Lines - MS. MAHER'S MATHmahersmath.weebly.com/.../tangent_lines_20180409134742.pdfMath 3 6.7 Tangent Lines of Circles Unit 6 SWBAT solve for unknown variables uslng theorems
Documents ##### Circles – Tangent Lines A tangent line touches a circle at exactly one point. In this case, line t is tangent to circle A. t A
Documents ##### How do I identify segments and lines related to circles? How do I use properties of a tangent to a circle?
Documents Documents