Circles

Question 1

Difference between two circles

Answer 1

square root (x2-x1)^2 + (y2-y1)^2

Question 2

Equation of a circle
(basic)

Answer 2

x^2 + y^2 = r^2
centre = (0,0)
radius = r

Question 3

Equation of a circle
(general)

Answer 3

(x-a)^2 + (y-b)^2 = r^2
centre = (a,b)
radius = r

Question 4

Equation of a circle
(extended form)

Answer 4

x^2 + y^2 + 2gx + 2fy + c = 0

centre = (-g,-f)
radius = square root g^2 + f^2 - c

Question 5

How to find the centre of a circle and radius

e.g x^2 + y^2 - 16x + 6y - 7 = 0

Answer 5

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

Question 6

How can you check things with a circle?

Is it a circle?

e.g x^2 + y^2 - 6x + 10y + 35 = 0

Answer 6

• 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.

Question 7

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

Answer 7

• 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.

Question 8

Circles

If the LHS > RHS

Answer 8

the point is outside the circle

Question 9

Circles

If the LHS = RHS

Answer 9

the point is on the circle

Question 10

Circles

If the LHS < RHS

Answer 10

the point is in the circle

Question 11

Equation of a tangent to a circle

facts

Answer 11

m1 x m2 = -1

y - b = m(x - a)

Question 12

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

Answer 12

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

Question 13

Intersection of a line and a circle

if it has two points of contact

Answer 13

b^2 - 4ac > 0

Question 14

Intersection of a line and a circle

If it has one point of contact (tangent)

Answer 14

b^2 - 4ac = 0

Question 15

Intersection of a line and a circle

If it has no points of contact

Answer 15

b^2 - 4ac < 0

Question 16

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

Answer 16

• 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.

Question 17

Intersection of two circles

If the intersect at two points

Answer 17

R1 + R2 > d

Question 18

Intersection of two circles

If they intersect at one point (tangent)

Answer 18

R1 + R2 = d

Question 19

Intersection of two circles

If they don’t intersect at all

Answer 19

R1 + R2 < d

Question 20

Intersection of two circles

What do you need to answer the question?

Answer 20

• centre1 + centre2
• radius1 + radius2
• distance between centres
  square root (x2 - x1)^2 + (y2 - y1)^2

Question 21

Intersection of two circles

Do they intersect?

e.g (x + 1)^2 + (y + 3)^2 = 4
x^2 + y^2 - 4x - 2y + 1 = 0

Answer 21

• 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.

Question 22

What does collinear mean?

Answer 22

three or more points on the same line
- check gradient & share a point

Question 23

What does concentric mean?

Answer 23

circles that share the same centre

Question 24

What does concurrent mean?

Answer 24

three or more lines intersect

Question 25

What does congruent mean?

Answer 25

same shape and size, but a different place