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Intro to Conics - Circles
I.. Parts of a Circle.
A) Standard form of a circle equation: x2 + y2 = r2
1) In a later section we will write the equation like this…
(x – h)2 + (y – k)2 = r2
B) The center is at ( 0 , 0 ).
1) If in (x – h)2 + (y – k)2 = r2 form, the center is (h , k).
2) Set the inside = 0 and solve (change the sign).
C) The radius of the circle can be found by taking the square
root of the # on the end ( r 2 = r ).
Intro to Conics - Circles
II.. Writing Equations of Circles in Standard Form x2 + y2 = #.
A) Get the x & y values on one side of the equal sign.
1) Move the # part to the other side of the equal sign.
B) Divide both sides by the value in front of the x2 (if needed).
Examples: Write the equation for a circle in standard form.
1) 6x2 – 30 = – 6y2
2) 8y2 + 38 = – 8x2 + 62
6x2 + 6y2 = 30
8x2 + 8y2 = 24
(divide by 6)
(divide by 8)
x2 + y2 = 5
x2 + y2 = 3
Intro to Conics - Circles
III.. Writing Equations of Circles from Points on the Circle.
A) If you know the center (0 , 0) and any point on the graph,
you can find the radius by using the distance formula.
1) Distance =
( x  x) 2  ( y  y ) 2
2) If the center is (0 , 0), then you have (x – 0)2 = x2 and
(y – 0)2 = y2. So the distance = ( x) 2  ( y ) 2
3) But the distance is the radius. The circle equation says
x2 + y2 = r2, so we have to square the distance.
a)
r2
= ( x)  ( y )
2
2
2
so
B) Now write the equation: x2 + y2 = #
r2 = (x)2 + (y)2.
Intro to Conics - Circles
IV.. Writing Equations of Circles given the Center & the Radius.
A) If the center is at the origin (0 , 0), then the standard
equation is x2 + y2 = #.
1) If the center is NOT at the origin, then you change the
signs to put it in (x – h)2 + (y – k)2 = # form.
Example: Center is (5 , – 8), then eq: (x – 5)2 + (y + 8)2 = #
B) If the center is at the origin (0 ,0) and you are told the value
of r, then square that value to get r2 (write it twice).
Example:
r = 8, so 82 = 64
x2 + y2 = 64
r = 3 5 , so (3 5 )  3  5
2
x2 + y2 = 45
2
2
Intro to Conics - Circles
V.. Graphing a Circle that is written in Standard Form.
A) Find and graph the center of the circle ( 0 , 0 ).
1) If in (x – h)2 + (y – k)2 = # form, the center is (h , k).
B) Find the radius of the circle.
1) Square root the end number.
C) Graph the “vertices” of the circle.
1) From the center, graph a dot that has a length of “r”
up, down, left and right of the center point.
2) Connect these four points with a round line.
HW: Circles
# 3 – 8 all, 9 – 19 odd, 23 – 29 odd, 45 – 51 odd