Circles Flashcards
Difference between two circles
square root (x2-x1)^2 + (y2-y1)^2
Equation of a circle
(basic)
x^2 + y^2 = r^2
centre = (0,0)
radius = r
Equation of a circle
(general)
(x-a)^2 + (y-b)^2 = r^2
centre = (a,b)
radius = r
Equation of a circle
(extended form)
x^2 + y^2 + 2gx + 2fy + c = 0
centre = (-g,-f)
radius = square root g^2 + f^2 - c
How to find the centre of a circle and radius
e.g x^2 + y^2 - 16x + 6y - 7 = 0
centre = 2g = -16 2f = 6
g = -8 f = 3
(8,-3)
radius = square root g^2 + f^2 - c
square root (-8)^2 + (3)^2 - (-7)
square root 64 + 9 + 7
square root 80
How can you check things with a circle?
Is it a circle?
e.g x^2 + y^2 - 6x + 10y + 35 = 0
- check radius & centre
2g = -6 2f = 10
g = -3 f = 5
(3,-5)
square root (-3)^2 + (5)^2 - 35
square root -1
Since the radius is not real, the equation is not a circle.
How to check things with a circle?
Does this point lie on a circle?
e.g (5,-2) (x - 4)^2 + (y + 9)^2 = 45
- sub co-ord into the LHS
(5 - 4)^2 + ((-2) +9)
1^2 + 7^2
50
Since the LHS does not equal the RHS, it does not lie on the circle.
Circles
If the LHS > RHS
the point is outside the circle
Circles
If the LHS = RHS
the point is on the circle
Circles
If the LHS < RHS
the point is in the circle
Equation of a tangent to a circle
facts
m1 x m2 = -1
y - b = m(x - a)
Equation of a tangent to a circle
e.g - the point A (8,6) lies on the circle
x^2 + y^2 - 4x - 6y - 104 = 0
centre - 2g = -4 2f = -6
g = -2 f = -3
(2,3)
gradient - m = y2 - y1/x2 - x1
= 6-3/8-2
= 1/2
tangent = m1 x m2 = -1
therefore = -2
equation
y - 6 = -2(x - 8)
y = -2x + 22
Intersection of a line and a circle
if it has two points of contact
b^2 - 4ac > 0
Intersection of a line and a circle
If it has one point of contact (tangent)
b^2 - 4ac = 0
Intersection of a line and a circle
If it has no points of contact
b^2 - 4ac < 0
Intersection of a line and a circle
e.g Determine the nature of the line
y = 2x - 3
and the circle (x - 3)^2 + (y + 2)^2 = 14
- sub y = 2x -3 into the circle
(x - 3)^2 + ((2x - 3) + 2)^2 = 14
x^2 - 6x + 9 + 4x^2 - 4x - 14 = 0
5x^2 - 10x - 4 = 0 - check discriminant
b^2 - 4ac
(-10)^2 - 4(5)(-4)
180
Statement - Since b^2 - 4ac > 0, there are two distinct points of intersection.
Intersection of two circles
If the intersect at two points
R1 + R2 > d
Intersection of two circles
If they intersect at one point (tangent)
R1 + R2 = d
Intersection of two circles
If they don’t intersect at all
R1 + R2 < d
Intersection of two circles
What do you need to answer the question?
- centre1 + centre2
- radius1 + radius2
- distance between centres
square root (x2 - x1)^2 + (y2 - y1)^2
Intersection of two circles
Do they intersect?
e.g (x + 1)^2 + (y + 3)^2 = 4
x^2 + y^2 - 4x - 2y + 1 = 0
- Find the centre & radius of each circle
C1 = (-1,-3) R1 = 2
C2 = 2g = -4 2f = -2
g = -2 f = -1
R2 = square root (-2)^2 + (-1)^2 -1
= 2
- Find the distance between the two circles
square root (2 - (-1))^2 + (1 - (-3))^2
square root 3^2 + 2^2
square root 9 - 4
square root 5 - Statement
Since R1 + R2 < 5, they do not intersect.
What does collinear mean?
three or more points on the same line
- check gradient & share a point
What does concentric mean?
circles that share the same centre
What does concurrent mean?
three or more lines intersect
What does congruent mean?
same shape and size, but a different place
