P.3 Circles & Symmetry. Symmetric to the y-axis ◦ Replace all of the x with –x Symmetric to the...
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Transcript of P.3 Circles & Symmetry. Symmetric to the y-axis ◦ Replace all of the x with –x Symmetric to the...
P.3 Circles& Symmetry
Symmetric to the y-axis◦ Replace all of the x with –x
• Symmetric to the x-axis– Replace all of the y with –y
• Symmetric to the origin– Replace all of the y with –y and x with -x
Check for symmetry with respect to both axes and the origin.
xy2 + 10 = 0
Answer– X-axis symmetry only
Check for symmetry with respect to both axes and the origin.
y = 9 – x2
Answer– Y-axis symmetry only
Check for symmetry with respect to both axes and the origin.
xy = 4
Answer– origin symmetry only
Equation of a Circle
Standard Form:
General Form:
(x – h)2 + (y – k) 2 = r2
Ax2 + Ay2 + Bx + Cy + D = 0
Standard Form:(x – h)2 + (y – k) 2 = r2
(h,k) = centerr = radius
(h,k)r
Determine the center and the radius of the following circles in standard form:
Equation Center
Radius
x2 + (y-5) 2 = 32
(x–1)2 + (y+7) 2 = 25
(x–2)2 + (y–3) 2 = 16
(x–1/3)2 + y 2 = 9/2
(x+6)2 +(x+2.3)2 =2.5
Determine the standard form of the following circles if given the center and the radius:
Equation Center
Radius
(-1,-5) r = 3
(3,4) r=8
(1.3,-6.5) r = 2.2
(0, -4) r =√ 3
(2,0) r = 5√ 5
General Form:Ax2 + Ay2 + Bx + Cy + D = 0
All circles in standard form can be easily converted to
general form…
how?
A,B,C & D are integers
Standard Form General Form
(x–2)2 + (y–3) 2 = 16
(x–1)2 + (y+7) 2 = 25
(x–1/3)2 + y 2 = 9/2
(x+6)2 +(y+2.3)2 =2.5
Determine the center and the radius of the following circle in general form:
x2 + y2 - 6x – 8y – 75 = 0
What do you think?
x2 + y2 - 6x – 8y – 75 = 0Divide every term by “A”
Group x’s and y’s…Move “D” to the other side of the =
(x2 - 6x ) + (y2 – 8y )= 75
Complete the square of both groups…Remember, whatever you add to the left,
be sure to add to the right.
(x2 - 6x + 9)+(y2 – 8y +16)=75+9+ 16
Factor each group and simplify the right.
(x – 3)2+(y – 4)2 =100
Now you can identify the center and the radius…
Determine the center and the radius of the following circles in general form:
Equation Center
Radius
x2 + y2 + 12x – 6y – 4 = 0
2x2 + 2y2 + 8x + 20y + 10 = 03x2 + 3y2 + 3x – 36y = 0
x2 + y2 – 14y – 1= 016x2 + 16y2 + 48x – 88y – 3 = 0
More ExamplesDetermine the standard form and general form of the following circle:
Center = (5,3) passing through the point (2,7)
(5,3)
(2,7)
Picture not drawn to scale
More ExamplesDetermine the standard form and general form of the following circle:
Endpoints of the diameter : (4,6) and (-8,1)
(-8,1)
(4,6)
Picture not drawn to scale