S56 (5.3) The Circle.notebook August 27, 2015

Daily Practice 19.8.2015

Q1. Calculate the value of a house initially worth £172 000 and increases in value every year by 2.3% for 5 years

Q2. State the gradient of the equation 2x - y = 14

Q3. State the angle that the line y = 4x - 1 makes with the positive direction of the x - axis

Q4. Write 3x2 - 12x + 6 in completed square form

Today we will be learning about

the equation of a circle.

The equation of a circle with centre (0,0)

(x, y)

r

(0, 0)

Given the centre of a circle is

(0, 0)

Write x and y in terms of r

x

y

The equation of a circle

The equation of a circle with centre (0,0)

is always x2 + y2 = r2

Examples:

1. State the equation of a circle with centre (0, 0)

and radius 3

2. State the equation of the circle that has centre origin and passes through (3, 4)

The equation of a circle

To show a point is on a circle,

substitute the point into the equation

of the circle.

Examples:

1. Show that the point (1, -1) is

on the circle x2 + y2 = 2

The equation of a circle

3. State whether the point (3, 3) lies inside or outside the circle with equation x2 + y2 = 16

Page 207. Ex. 12D

Q1, 2, 3b, c, e, Q4

Q5, 6, Q8 a, d

Q10.

S56 (5.3) The Circle.notebook August 27, 2015

Daily Practice 20.8.15

Q1. State the radius of the circle with equation x2 + y2 = 144

Q2. State the equation of the altitude from B of the triangle A(6, 3)

B(1, 2) and C(4, 11)

Q4. The roots of the equation kx2 ‐ 3x + 2 = 0 are equal, what is the

value of k?

Today we will be working out the equations of

circles with centres that are not the origin.

The equation of a circle (Standard Form)

1 2 3 4 5-1-2-3-4-5

-1

-2

-3

-4

-5

1

2

3

4

x

y 5

(x, y)

r

Writing x and y in terms of r

when the centre isn't (0,0)

(a, b)

The equation of a circle (Standard Form)

So the standard equation of a circle is (x - a)2 + (y - b)2 = r2 where

(a, b) is the centre and r is the radius

Examples:

1. State the equation of a circle with centre (3, 2) and radius 2√3

2. State the radius and the centre of the circle with equation

(x - 7)2 + (y + 3)2 = 36

The equation of a circle (Standard Form)

3. The centre of a circle is (-1, 8) and the circle passes through

(-1, 16). Calculate the radius and hence find the equation of the circle.

Page 210, Ex. 12F

Q1 a, d, e Q2 b, d

Q3 - 8 Q10 a, d, f

Daily Practice 21.8.15

Q1.

S56 (5.3) The Circle.notebook August 27, 2015

The general equation of a circle

Given a circle in the form (x - a)2 + (y - b)2 = r2 , we can multiply out and simplify to get the expanded form

Example: Write without brackets in its simplest form

Ex. 12G Q1.

(x - 8)2 + (y - 3)2 = 100

Today we will be writing the equation of the

circle in expanded form (the general equation)

The equation of a circle: General Form

Example: Given this equation of a circle in expanded form, find the

centre and the radius

x2 + y2 - 10x - 6y - 2 = 0

The equation of a circle: General Form

The equation of a circle: General Form

The general form of the equation of a circle is just the multiplied out version of the standard form.

+ 2gx + 2fy + c = 0

(-g, -f) and radius g

The circle will exist if it has a radius i.e. r > 0 and the coefficients of x2 and y2 are the same.

Hence circle exists

The equation of a circle: General Form

Examples:

1. Write down the centre and the radius of the circle

x2 + y2 + 2x + 4y - 27 = 0

2. Write down the centre and the radius of the circle

x2 + y2 - 6x - 2y - 30 = 0

S56 (5.3) The Circle.notebook August 27, 2015

The equation of a circle: General Form

3. Prove that x2 + y2 + 8x - 14y + 66 = 0 is not an equation of a circle

Pg. 213 (ii) (iii)

a, d, g, j, k, l

Q2, 4 b, d, Q8.

Daily Practice 24.8.2015

Q1. State the centre and radius of a circle with equation

(x - 1)2 + (y + 3)2 = 48

Q2. State the gradient of the line shown

Q3. Write 5x2 - 15x + 10 in completed square form

Q4. Given that 2x2 + px + p + 6 = 0 has equal roots, find the possible

values for p

1380

Today we will be finding the point of

intersection of a line and a circle.

Homework Online due 1.9.2015

Intersection of a line and a circle

To find the point(s) of intersection of

a circle and a line, substitute the equation

of the line into the equation of the circle.

Example: Find where the line y = x + 5

intersects the circle x2 + y2 + 4x - 6y + 5 = 0

Pg. 217 Q1 (ii), a, f

Q2. (ii) b

Q3.

Daily Practice 25.8.2015

Q1. Write 5x2 - 10x + 8 in completed square form

Q2.

S56 (5.3) The Circle.notebook August 27, 2015

Today we will be understanding and working with tangents

to circles.

Homework Due Tuesday.

Intersection of a line and a circle

Intersection of a line and a circle

We can use the discriminant to show the point(s) of intersection, if there are any of a line and a circle.

b2 - 4ac > 0 means that there are 2 different points of intersection

b2 - 4ac = 0 means that there is one point of intersection i.e. a tangent

b2 - 4ac < 0 means that there are no points of intersection.

Intersection of a line and a circle

If there is only one point of intersection,

then the line is a tangent to the circle.

Examples:

1. Show that the line 2y - x = 9

is tangent to the circle x2 + y2 + 2x - 2y -18 = 0 and

find the point of contact

Daily Practice 26.8.15

Q1. State the centre and radius of the circle x2 + y2 + 18x ‐ 8y ‐ 143 = 0

Q2. State the equation of the perpendicular bisector of the line

joining A(1, ‐2) and B(3, 2)

Q3. State the nature of the roots of the function f(x) = ‐5x2 + 2x ‐ 3

S56 (5.3) The Circle.notebook August 27, 2015

Today we will be continuing to work with tangents to

circles.

Homework Due Tuesday.

Tangent to a circle

Examples:

2. Find the equation of the tangent to the circle with equation

x2 + y2 + 2x + 4y - 27 = 0 where the point of contact P is P(3, 2)

Page 223 Ex. 12K

Q3,6 (b), (c)

Ex. 12 L Q2, 3

Example

Show that the line y - 2x + 4 = 0 is a tangent to the circle - 2x - 6y + 5 = 0 and find the point of contact.

DailyPractice 27.8.2015

Q1. State the centre and radius of the circle x2 + y2 - 4x - 6y = 413

Q2. Find the equation of the perpendicular bisector of the line

joining A(2, 4) and B(8, 6)

Q3. The roots of the equation y = kx2 - 3x + 2 = 0 are equal, state the

value of k

Q5. Sketch a parabola of the form y = ax2 + bx + c where b2 = 4ac > 0

Today we will be learning about circles touching.

S56 (5.3) The Circle.notebook August 27, 2015

Circles touching internally/externally

If 2 circles touch externally, then the distance between their centres is equal to the sum of their radii.

d = r1 + r2

If 2 circles touch internally, then the distance between their centres is equal to the difference of their radii.

d = r1 - r2

r2 r1

d

d r2

r1

Circles touching internally/externally

d = r1 + r2 d = r1 - r2

d > r1 + r2 d = 0

r1 > r2

d < r1 - r2

Example:

Daily Practice 28.8.2015

Q1. State the gradient of the line 3y - 2x + 4 = 0

Q2. State the size of the angle that the line 0.5x + 3 = y makes

with the positive direction of the x axis

Q3. Write 2x2 + 4x + 5 in completed square form

Q4. State the radius and centre of the circle

(x - 3)2 + (y + 4)2 = 72

Today we will be practising mixed questions on the circle.

Homework Due Tuesday.

Scholar Passwords

1.

2.

3.

S56 (5.3) The Circle.notebook August 27, 2015

The Circle

Concentric circles are circles that are within each other (have the same centre)

The equation of a circle

with centre (0, 0) & radius r

is x2 + y2 = r2

The equation of a circle

with centre (a, b) & radius r

is (x - a)2 + (y - b)2 = r2

The equation of a

with centre (-g, -f) & radius r

is + 2gx + 2fy + c = 0 where r = g2 + f2 - c

If a line intersects a circle at 2 points it is a chord

If a line intersects a circle at 1 point it is a tangent