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Higher. Circle. Unit 2 Outcome 4. The General equation of a circle. x 2 + y 2 + 2gx + 2fy + c = 0. Wednesday, 07 January 2009. Higher. Circle. Unit 2 Outcome 4. x 2 + y 2 + 2gx + 2fy + c = 0. In the same way we can. The equation of a circle is - PowerPoint PPT PresentationTRANSCRIPT
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Higher Unit 2 Outcome 4Circle
x 2 + y 2 + 2gx + 2fy + c = 0
The General equation of a circle
Wednesday, 07 January 2009
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Higher Circle Unit 2 Outcome 4
x 2 + y 2 + 2gx + 2fy + c = 0 The equation of a circle is
(x – 2)2 + (y – 3)2 = 25
Write the equation without brackets
(x – 2) (x – 2) + (y – 3) (y – 3) = 25 x2 - 4x + 4 + y2 - 6y + 9 = 25
x2 - 4x + y2 - 6y + 13 - 25 = 0
x2 + y2 - 4x - 6y - 12 = 0
x2 + y2 – 2ax – 2by +a2 +b2 – r2 = 0
(x – a) (x – a) + (y – b) (y – b) = r2
(x – a) 2 + (y – b) 2 = r2
x2 – 2ax + a2 + y2 - 2by + b2 = r2
As a , b and r are constants (numbers) then these can be collected together as one term, c
x2 + y2 – 2ax – 2by + c = 0
In the same way we can
This is the general form This is the general form of the equation of a circleof the equation of a circle
Wednesday, 07 January 2009
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222 )()( rbyax Centre C(a,b) Radius r1.
Radius r02222 cfygxyx Centre C(-g,-f) cfg 222.
222 rba c -b, f a,- g Let
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x 2 + y 2 + 2gx + 2fy + c = 0 Higher Unit 2 Outcome 4Circle
Wednesday, 07 January 2009
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Finding the centre and the radiusFinding the centre and the radius
Given the equation of a circle, we can find the coordinates of its centre and the length of its radius. For example:
Find the centre and the radius of a circle with the equation (x – 2)2 + (y + 7)2 = 64
By comparing this to the general form of the equation of a circle of radius r centred on the point (a, b):
(x – a)2 + (y – b)2 = r2
We can deduce that for the circle with equation
(x – 2)2 + (y + 7)2 = 64
The centre is at the point (2, –7) and the radius is 8.
Wednesday, 07 January 2009
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Finding the centre and the radiusFinding the centre and the radius
When the equation of a circle is given in the form
Find the centre and the radius of a circle with the equation x2 + y2 + 4x – 6y + 9 = 0
Start by rearranging the equation so that the x terms and the y terms are together:
x2 + 4x + y2 – 6y + 9 = 0
x2 + y2 – 2ax – 2by + c = 0
we can use the method of completing the square to write it in the form
(x – a)2 + (y – b)2 = r2
For example:
Wednesday, 07 January 2009
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Finding the centre and the radiusFinding the centre and the radius
We can complete the square for the x terms and then for the y terms as follows:
The equation of the circle can now be written as:
x2 + 4x = (x + 2)2 – 4
y2 – 6y = (y – 3)2 – 9
(x + 2)2 – 4 + (y – 3)2 – 9 + 9 = 0
(x + 2)2 + (y – 3)2 = 4
(x + 2)2 + (y – 3)2 = 22
The centre is at the point (–2, 3) and the radius is 2.
x2 + 4x + y2 – 6y + 9 = 0
Wednesday, 07 January 2009
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Higher Circle Unit 2 Outcome 4
x 2 + y 2 + 2gx + 2fy + c = 0 Alternative approach
Rearrange to get in the general form
x2 + 4x + y2 – 6y + 9 = 0
x2 + y2 + 4x – 6y + 9 = 0
2g = 4 2f = -6 c = 9
x 2 + y 2 + 2gx + 2fy + c = 0
g = 2 f = -3 c = 9
As before It therefore follows that
The centre is at the point (–2, 3) and the radius is 2.
(x + 2)2 + (y – 3)2 = 22
C is sum of all the constants
Wednesday, 07 January 2009
r2 = g2 +f2 - c
r2 = 22 + - 32 - 9
Centre (-g, -f)
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Higher Circle Unit 2 Outcome 4
x 2 + y 2 + 2gx + 2fy + c = 0
Wednesday, 06 January 2009
Show that the equation x2 + y2 - 6x + 2y - 71 = 0represents a circle and find the centre and radius.
x2 + y2 - 6x + 2y - 71 = 02g = -6 2f = 2 c = -71
g = -3 f = 1 c = -71
(x + 3)2 + (y – 1)2 = 92
r2 = g2 + f2 -c
r2 = 9 + 1 - -71
r2 = 81
This is now in the form (x-a)2 + (y-b)2 = r2
So represents a circle with centre (3,-1) and radius = 9
Centre (-g, -f)
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Higher Circle Unit 2 Outcome 4
x 2 + y 2 + 2gx + 2fy + c = 0
Wednesday, 06 January 2009
Show that the equation x2 + y2 + 6x - 2y - 15 = 0represents a circle and find the centre and radius.
x2 + y2 + 6x - 2y - 15 = 02g = 6 2f = -2 c = -15
g = 3 f = -1 c = -15
(x - 3)2 + (y + 1)2 = 52
r2 = g2 + f2 -c
r2 = 9 + 1 - -15
r2 = 25
This is now in the form (x-a)2 + (y-b)2 = r2
So represents a circle with centre (-3,1) and radius = 5
Centre (-g, -f)
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Higher Circle Unit 2 Outcome 4
x 2 + y 2 + 2gx + 2fy + c = 0
Wednesday, 06 January 2009
Show that the equation x2 + y2 - 4x - 6y + 9 = 0represents a circle and find the centre and radius.
x2 + y2 - 4x - 6y + 9 = 02g = -4 2f = -6 c = 9
g = -2 f = -3 c = 9
(x + 2)2 + (y + 3)2 = 22
r2 = g2 + f2 -c
r2 = 4 + 9 - 9
r2 = 4
This is now in the form (x-a)2 + (y-b)2 = r2
So represents a circle with centre (2,3) and radius = 2
Centre (-g, -f)
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Higher Circle Unit 2 Outcome 4
x 2 + y 2 + 2gx + 2fy + c = 0
Wednesday, 06 January 2009
Show that the equation x2 + y2 + 2x + 8y + 1 = 0represents a circle and find the centre and radius.
x2 + y2 + 2x + 8y + 1 = 02g = 2 2f = 8 c = 1
g = 1 f = 4 c = 1
(x - 1)2 + (y - 4)2 = 42
r2 = g2 + f2 - c
r2 = 1 + 16 -1
r2 = 16
This is now in the form (x-a)2 + (y-b)2 = r2
So represents a circle with centre (-1,-4) and radius = 4
Centre (-g, -f)
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Higher Unit 2 Outcome 4Circle
Wednesday, 06 January 2009
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To build skills Complete
Exercise 3A Q 1, Q2,
(x – a)2 + (y – b)2 = r2 Centre C (a,b)(a,b) and radius rr
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Higher Unit 2 Outcome 4Circle
Tuesday, 06 January 2009
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