Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4....
Transcript of Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4....
![Page 1: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/1.jpg)
Section 1.4
Circles
Math 1051 - Precalculus I
Circles Section 1.4
![Page 2: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/2.jpg)
Section 1.4 Circles
Find the equation of the line that passes through (2,−3) and(1,−2)
Circles Section 1.4
![Page 3: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/3.jpg)
From last time...
What can you say about 2 lines that have the same x-interceptand the same y -intercept?
Circles Section 1.4
![Page 4: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/4.jpg)
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Circles Section 1.4
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Circles Section 1.4
![Page 6: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/6.jpg)
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Circles Section 1.4
![Page 7: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/7.jpg)
Circle
How to describe a circle:
Start with a fixed point (h, k)Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)
Definitions:r is the radius(h, k) is the center
Circles Section 1.4
![Page 8: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/8.jpg)
Circle
How to describe a circle:Start with a fixed point (h, k)
Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)
Definitions:r is the radius(h, k) is the center
Circles Section 1.4
![Page 9: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/9.jpg)
Circle
How to describe a circle:Start with a fixed point (h, k)Give a distance r
A circle is all the points in the xy -plane that are a distancer from the point (h, k)
Definitions:r is the radius(h, k) is the center
Circles Section 1.4
![Page 10: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/10.jpg)
Circle
How to describe a circle:Start with a fixed point (h, k)Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)
Definitions:r is the radius(h, k) is the center
Circles Section 1.4
![Page 11: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/11.jpg)
Circle
How to describe a circle:Start with a fixed point (h, k)Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)
Definitions:r is the radius(h, k) is the center
Circles Section 1.4
![Page 12: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/12.jpg)
Circle
How to describe a circle:Start with a fixed point (h, k)Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)
Definitions:r is the radius(h, k) is the center
Circles Section 1.4
![Page 13: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/13.jpg)
Circles Section 1.4
![Page 14: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/14.jpg)
Facts about circles
A = πr2
c = 2πrπ ≈ 3.1415926535897932384626433832795π is an irrational numberπ is a transcendental numberBut
π = 4∞∑
k=0
(−1)k
2k + 1=
41− 4
3+
45− 4
7+ · · ·
Circles Section 1.4
![Page 15: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/15.jpg)
Facts about circles
A = πr2
c = 2πr
π ≈ 3.1415926535897932384626433832795π is an irrational numberπ is a transcendental numberBut
π = 4∞∑
k=0
(−1)k
2k + 1=
41− 4
3+
45− 4
7+ · · ·
Circles Section 1.4
![Page 16: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/16.jpg)
Facts about circles
A = πr2
c = 2πrπ ≈ 3.1415926535897932384626433832795
π is an irrational numberπ is a transcendental numberBut
π = 4∞∑
k=0
(−1)k
2k + 1=
41− 4
3+
45− 4
7+ · · ·
Circles Section 1.4
![Page 17: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/17.jpg)
Facts about circles
A = πr2
c = 2πrπ ≈ 3.1415926535897932384626433832795π is an irrational number
π is a transcendental numberBut
π = 4∞∑
k=0
(−1)k
2k + 1=
41− 4
3+
45− 4
7+ · · ·
Circles Section 1.4
![Page 18: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/18.jpg)
Facts about circles
A = πr2
c = 2πrπ ≈ 3.1415926535897932384626433832795π is an irrational numberπ is a transcendental number
But
π = 4∞∑
k=0
(−1)k
2k + 1=
41− 4
3+
45− 4
7+ · · ·
Circles Section 1.4
![Page 19: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/19.jpg)
Facts about circles
A = πr2
c = 2πrπ ≈ 3.1415926535897932384626433832795π is an irrational numberπ is a transcendental numberBut
π = 4∞∑
k=0
(−1)k
2k + 1=
41− 4
3+
45− 4
7+ · · ·
Circles Section 1.4
![Page 20: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/20.jpg)
Great Pyramid of GizaHas a perimeter of 1760 cubits and a height of 280 cubits
1760280
≈ 2π
Circles Section 1.4
![Page 21: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/21.jpg)
Great Pyramid of GizaHas a perimeter of 1760 cubits and a height of 280 cubits
1760280
≈ 2π
Circles Section 1.4
![Page 22: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/22.jpg)
Great Pyramid of GizaHas a perimeter of 1760 cubits and a height of 280 cubits
1760280
≈ 2π
Circles Section 1.4
![Page 23: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/23.jpg)
Formula for a circle
Standard form
(x − h)2 + (y − k)2 = r2
General form
x2 + y2 + ax + by + c = 0
Circles Section 1.4
![Page 24: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/24.jpg)
Formula for a circle
Standard form
(x − h)2 + (y − k)2 = r2
General form
x2 + y2 + ax + by + c = 0
Circles Section 1.4
![Page 25: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/25.jpg)
Formula for a circle
Standard form
(x − h)2 + (y − k)2 = r2
General form
x2 + y2 + ax + by + c = 0
Circles Section 1.4
![Page 26: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/26.jpg)
Examples
Find the center, radius, and intercepts of
3(x + 1)2 + 3(y − 1)2 = 6
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Circles Section 1.4
![Page 27: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/27.jpg)
Examples
Find the center, radius, and intercepts of
3(x + 1)2 + 3(y − 1)2 = 6
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Circles Section 1.4
![Page 28: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/28.jpg)
Examples
Find the center, radius, and intercepts of
x2 + y2 + 4x + 2y − 20 = 0
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Circles Section 1.4
![Page 29: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/29.jpg)
Examples
Find the center, radius, and intercepts of
x2 + y2 + 4x + 2y − 20 = 0
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-10
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0
5
10
Circles Section 1.4
![Page 30: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/30.jpg)
Examples
Find the center, radius and intercepts of
2x2 + 2y2 + 8x + 7 = 0
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0
2
4
Circles Section 1.4
![Page 31: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/31.jpg)
Examples
Find the center, radius and intercepts of
2x2 + 2y2 + 8x + 7 = 0
-4 -2 0 2 4
-4
-2
0
2
4
Circles Section 1.4
![Page 32: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/32.jpg)
Examples
Find the equation of a circle with endpoints of a diameter (4,3)and (0,1)
Important: The information we need for a circle is the centerand radius
Circles Section 1.4
![Page 33: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/33.jpg)
Examples
Find the equation of a circle with endpoints of a diameter (4,3)and (0,1)
Important: The information we need for a circle is the centerand radius
Circles Section 1.4
![Page 34: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/34.jpg)
Examples
Find the equation of a circle with center (4,−2) and tangent tothe line x = 1.
Circles Section 1.4
![Page 35: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/35.jpg)
Examples
Find the area of the region between the circle x2 + y2 = 36 andthe inscribed square.
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Circles Section 1.4
![Page 36: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/36.jpg)
Examples
Find the area of the region between the circle x2 + y2 = 36 andthe inscribed square.
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0
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10
Circles Section 1.4
![Page 37: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/37.jpg)
Examples
Given the circle x2 + y2 = r2 and a line y = mx + b tangent tothe circle, find b as a function of r and m.
Circles Section 1.4
![Page 38: Section 1 - University of Minnesotahankx003/Fall2012/Lectures/Ch1Sec4.pdf · Circles Section 1.4. Circle How to describe a circle: Start with a fixed point (h;k) Give a distance](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fccc53e6ad0c667386c680e/html5/thumbnails/38.jpg)
Read sections 2.1 and 2.2 for Monday
Circles Section 1.4