Y1, C6 - Circles

Question 1

How do you find the perpendicular bisector of two points

Answer 1

Find their midpoint
Find the gradient using the negative reciprocal of the gradient between the two points
Sub in x and y values to find c

Question 2

What is the equation for a circle of radius r with centre at origin

Answer 2

x^2 + y^2 = r^2

Question 3

What is the equation for a circle of radius r with centre at (a, -b)

Answer 3

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

Question 4

Find the centre and radius of the circle with equation x^2 + y^2 - 6x + 2y - 7 = 0

Answer 4

Complete square for x’s and y’s
(x - 3)^2 - 9 + (y + 1)^2 - 1 - 6 = 0
(x - 3)^2 + (y + 1)^2 = 16
Centre = (3, -1)
Radius = 4

Question 5

Circle with equation x^2 + y^2 - 4x + 10y = k, state possible range of values for k

Answer 5

(x - 2)^2 + (y + 5)^2 - 29 = k
radius (k) cannot = 0
therefore k > -29

Question 6

How would you show if a straight line intersects with a circle once (tangent), two times (secant) or none

Answer 6

1) Solve simultaneously
2) Produce a quadratic
3) Check discriminant:
0 = 1 (tangent)
> 0 = 2
< 0 = none

Question 7

Show y = x + 3 never intersects the circle x^2 + y^2 = 1

Answer 7

x^2 (x + 3)^2 = 1
2x^2 + 6x + 8 = 0
b^2 - 4ac = -28 < 0, hence 0 solutions

Question 8

What does the perpendicular bisector of any chord pass through

Answer 8

The centre of the circle

Question 9

If you know the equation of a circle, how would you find the two equations for a tangent line ( l ) with the same gradient

Answer 9

1) Find equation of the line through the centre perpendicular to the tangents
2) Find the intersections of this line with the circle
3) Find the equations for l

Question 10

Circle passes through A(0, 0) and B(4, 2) centre has x-coordinate -1, what is the circle’s equation

Answer 10

Gradient of AB = 1/2
Gradient of perp bisector = -2
Midpoint of bisector = (2, 1)
Equation of bisector line = y = -2x + 5
as x = 1, y = 7
Centre of circle = (-1, 7)
Radius is from (-1, 7) to (0, 0) = root(50)
ans = (x+1)^2 + (y-7)^2 = 50

Question 11

What does it mean if a triangle inscribes a circle

Answer 11

It is inside and its vertices touch the circle without intersecting (crossing)

Question 12

What does it mean if a circle circumscribes a triangle (circumcircle)

Answer 12

The triangle is inscribing the circle

Question 13

What is the centre of a circumcircle called

Answer 13

Circumcentre

Question 14

If you know points A, B, and C on a circle, how do you show that AB is a diameter (2)

Answer 14

1) Show that AC^2 + BC^2 = AB^2
OR
2) Show that AC is perpendicular to BC

Question 15

Points on a circle: A(-8, 1), B(4, 5), C(-4, 9), show AB is a diameter

Answer 15

1) AC^2 + BC^2 = AB^2
AC^2 = 16 + 64 = 80
BC^2 = 64 + 16 = 80
AB^2 = 144 + 16 = 180
80 + 80 = 160 therefore Pythagoras holds
OR
2) Gradient AC = 2
Gradient BC = -1/2
M(AC) * M(BC) = -1
Therefore AC and BC are perpendicular thus AB is a diameter

Question 16

Given three points (a triangle in a circle), how can we find the centre of the circumcircle

Answer 16

Find the equation of the perpendicular bisectors for two different sides
Find their intersection point