S56 (5.1) The Circle.notebook November 23, 2016

Daily Practice 1.11.2016

Q1. The roots of the equation kx2 - 3x + 2 = 0 are equal, what is the value of k?

Q2. State the equation perpendicular to 2x - y = 14 that passes through (9, 1)

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

Daily Practice 2.11.2016

Q1. State the limit of the recurrence relation un+1 = 0.88un ‐ 4

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

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

Q3. The graph of

f(x) = 2x + 1 is shown.

State the inverse function f‐1(x) and

sketch the graph f‐1(x)

(0, 2)

Today we will be learning about

the equation of a circle.

Homework Quiz Friday

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

yThe equation of a circle

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)

S56 (5.1) The Circle.notebook November 23, 2016

The equation of a circle

To show a point is on a circle,

substitute the point into the equation

of the circle.

Examples:

3. 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

The equation of a circle

Questions:

1. State the equation of the circles with centre (0, 0) and

(i) radius = 6 (ii) r = 12 (iii) r = 2√3

(iv) passes through (6, 8) (v) passes through (-2, 1)

2. State the radius of the following circles:

(i) x2 + y2 = 49 (ii) x2 + y2 = 20 (iii) x2 + y2 = 72

3. State whether each point lies inside/outside or on the circle

x2 + y2 = 41

(i) (3, 7) (ii) (4, 5) (iii) (-1, 2) (iv) (2, 6)

Daily Practice 3.11.16

Q1. State the centre and radius of the circle x2 + y2 = 300

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

Q4. Factorise fully 2t3 + 5t2 ‐ 28t ‐ 15

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

Equation of the circle:

(a, b)

S56 (5.1) The Circle.notebook November 23, 2016

The equation of a circle (Standard Form)

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 4.11.2016 Daily Practice 7.11.2016

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 writing the equation of the

circle in expanded form (the general equation)

S56 (5.1) The Circle.notebook November 23, 2016

The general equation of a circle

State the centre and the radius and then without brackets in its simplest form

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

The equation of a circle: General Form

Reverse the process and then state the centre and the radius. (Try to express in completed square form)

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

The equation of a circle: General FormThe 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

+ 2gx + 2fy + c = 0

(-g, -f) and radius g

Daily Practice 8.11.2016

Today we will be finding the point of

intersection of a line and a circle.

S56 (5.1) The Circle.notebook November 23, 2016

The equation of a circle: General Form

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

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

Intersection of a line and a circle

Intersection of a line and a circle

Examples:

1. Find where the line y = x + 5

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

S56 (5.1) The Circle.notebook November 23, 2016

Daily Practice 9.11.2016

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

Q2.

Today we will be understanding and working with tangents

to circles.

Intersection of a line and a circle

2. Show that the line 2y - x = 9 is a tangent to the circle

x2 + y2 + 2x + 2y -18 = 0 and find the point of contact

1.

2.

3.

4.

++

++++

+

+

+ +

DailyPractice 10.11.2016

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 are equal, state the value of k

Q4. Sketch a parabola of the form y = ax2 + bx + c where b2 - 4ac > 0

S56 (5.1) The Circle.notebook November 23, 2016

Today we will be working out how to get the equation of a tangent to a circle.

Tangent to a circle

Examples:

State 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)

1.

2.

Daily Practice 14.11.2016

Q1. State the equation of the line perpendicular to 3y - 2x + 4 = 0

that passes through (1, 7)

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 learning about the intersection of

circles.

S56 (5.1) The Circle.notebook November 23, 2016

Intersection of Circles

1. Concentric circles - 2 circles that have the same centre.

2. Two circles that touch externally

3. Two circles that touch internally

4. Two circles that don't touch

5. Circles meet at 2 distinct points

*

We can prove that circles intersect by looking at the distance between their centres and their radii.Congruent means the exact same shape and size.

Example:

Daily Practice 15.11.2016

Today we will be practising mixed questions on the

circle.

S56 (5.1) The Circle.notebook November 23, 2016

Daily Practice 16.11.2016

Q1.

1.

2.

3.

4.

Daily Practice 17.11.2016

1.

2.