circles notes. 1 st day a circle is the set of all points p in a plane that are the same distance...
TRANSCRIPT
![Page 1: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/1.jpg)
Circles Notes
![Page 2: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/2.jpg)
1st Day
![Page 3: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/3.jpg)
A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius of the circle, and the given point is the center of the circle.
![Page 4: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/4.jpg)
Standard form of a circle with center C(h, k) and radius r is
The standard form of the equation of a circle with center at the origin (0, 0) and radius r is .
Standard Equation
2 2 2 x h y k r
2 2 2 x y r
![Page 5: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/5.jpg)
Express in standard form the equation of the circle centered at (−2, 3) with radius 5.
Example 1
![Page 6: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/6.jpg)
2 2 2
2 2 2
2 2
2 3 5
2 3 25
x h y k r
x y
x y
![Page 7: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/7.jpg)
Express in standard form the equation of the circle with center at the origin and radius of 4. Sketch the graph.
Example 2
2 2
2 2 2
2
2 2
4
16
x y
x y
x y r
![Page 8: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/8.jpg)
-8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
![Page 9: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/9.jpg)
Find the center and radius of the circle with the equation
Example 3
2 2( 5) ( 1) 16 x y
2 2
22 25 1 4
center 5, 1 ; radius 4
x h y k r
x y
![Page 10: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/10.jpg)
Find the center and radius of the circle with the given equation
To find the center and radius you must rewrite this equation into standard form by completing the square twice.
Example 4
2 2 4 6 5 0 x y x y
![Page 11: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/11.jpg)
2 2
2 2
22 2 2
2 2
2 2
4 6 5 0
4 ____ 6 ____ 5 ____ ____
4 2 6 3 5 4 9
4 4 6 9 8
2 3 8
center 2, 3 ; radius 8 2 2
x y x y
x x y y
x x y y
x x y y
x y
![Page 12: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/12.jpg)
Find the center and radius of the circle with the given equation
Example 5
2 2 6 10 4 0 x y x y
![Page 13: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/13.jpg)
2 2
2 2
22 2 2
2 2
2 2
6 10 4 0
6 ____ 10 ____ 4 ____ ____
6 3 10 5 4 9 25
6 9 10 25 38
3 5 38
center 3, 5 ; radius 38
x y x y
x x y y
x x y y
x x y y
x y
![Page 14: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/14.jpg)
The graph of x2 + y2 = 16 is translated 3 units to the right and 2 units down. Find the equation of the new circle and sketch its graph.
Example 6
22x 3 y 2 16
![Page 15: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/15.jpg)
-8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
![Page 16: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/16.jpg)
2nd Day
![Page 17: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/17.jpg)
Write the equation for each circle described.
Example 1
![Page 18: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/18.jpg)
a. The circle has its center at (8, -9) and passes through the point at (4, -6).
2 2 28 9 x x r
22 28 4 9 6 r216 9 25 r
2 28 9 25 x x
![Page 19: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/19.jpg)
b. The endpoints of a diameter are at (1, 8) and (1, −4).
8 41 1center = , 1, 6
2 2
2 22 1 1 8 6 0 4 4 r
2 21 6 4 x x
![Page 20: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/20.jpg)
The circle with center (3, 4) and tangent to the y–axis.
If a circle is tangent to a line, then a radius of the circle is perpendicular to the line.
r = 3
-8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
2 23 4 9 x y
![Page 21: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/21.jpg)
Find the coordinates of the center and the radius of
Do completing the square twice.
Example 2
2 22 12 2 4 12 x x y y
![Page 22: Circles Notes. 1 st Day A circle is the set of all points P in a plane that are the same distance from a given point. The given distance is the radius](https://reader031.vdocument.in/reader031/viewer/2022031712/56649dbb5503460f94aad5af/html5/thumbnails/22.jpg)
2 22 6 _ 2 2 _ 12 2 _ 2 _ x x y y
2 22 12 2 4 12 x x y y
2 22 6 9 2 2 1 12 2 9 2 1 x x y y
2 22 3 2 1 8 x y
2 23 1 4 x y
center = (3, 1); radius = 2