AS-level Mathematics
Academy of Global Curriculum © 2024
Lesson objectives:
•
Definition of a circle
•
Find the radius and centre of a
circle from the equation of a circle:
•
Standard form
•
Completed square form
•
Find the equation of a circle given
the radius and centre.
Academy of Global Curriculum © 2024
Definition: locus (all possible positions / set of) of all the points in a
plane which are a fixed distance (radius) from a given point (centre)
Pythagoras Theorem: c2 = a2 + b2
Academy of Global Curriculum © 2024
Definition: locus (all possible positions / set of) of all the points in a
plane which are a fixed distance (radius) from a given point (centre)
(x,
y)
Pythagoras Theorem: c2 = a2 + b2
(y)
Centre = (0,0)
Radius = 5
(5)
(x)
We cam use this information to find
the standard equation of a circle.
Academy of Global Curriculum © 2024
Definition: locus (all possible positions / set of) of all the points in a
plane which are a fixed distance (radius) from a given point (centre)
(2,3
)
Calculate the radius of this circle
given that the centre of the circle is at
the origin and that the point (2,3) lies
on the circle.
r2 = (x-0)2 + (y-0)2
Academy of Global Curriculum © 2024
Definition: locus (all possible positions / set of) of all the points in a
plane which are a fixed distance (radius) from a given point (centre)
(12,7.
65)
(9,5
)
What about when the centre is not at
the origin? Lets find the equation of
the circle.
r2 = (x-0)2 + (x-0)2
r2 = (x-9)2 + (x-5)2
r2 = (12-9)2 + (7,65-5)2
Academy of Global Curriculum © 2024
Definition: locus (all possible positions / set of) of all the points in a
plane which are a fixed distance (radius) from a given point (centre)
(r)
(a,b
)
Standard Formula:
Radius = r
Centre = (a,b)
r2 = (x-a)2 + (y-b)2
Academy of Global Curriculum © 2024
Definition: locus (all possible positions / set of) of all the points in a
plane which are a fixed distance (radius) from a given point (centre)
Standard Formula:
(r)
(a,b
)
r2 = (x-a)2 + (x-b)2
What happens if we expand the brackets..
Academy of Global Curriculum © 2024
Definition: locus (all possible positions / set of) of all the points in a
plane which are a fixed distance (radius) from a given point (centre)
Academy of Global Curriculum © 2024
Definition: locus (all possible positions / set of) of all the points in a
plane which are a fixed distance (radius) from a given point (centre)
Completed Square Formula:
(r)
0 = x2 + y2 -2ax -2bx + (a2 + b2 -r2 )
0 = x2 + y2 +2(g)x +2(f)x + c
(a,b
)
g = -a
f = -b
Centre = ( -g, -f)
Radius =
g2 + f2 −c )
Academy of Global Curriculum © 2024
c = (a2 + b2 -r2 )
Completed Square Formula:
0 = x2 + y2 -2ax -2bx + (a2 + b2 -r2 )
g = -a
f = -b
c = (a2 + b2 -r2 )
0 = x2 + y2 +2(g)x +2(f)x + c
Centre = ( -g, -f)
Radius =
g2 + f2 −c )
Find the centre and radius of a circle given x2 + y2 -6x +10y -15 = 0
Academy of Global Curriculum © 2024
Academy of Global Curriculum © 2024