s56 (5.1) the circle.notebook november 23, 2016€¦ · s56 (5.1) the circle.notebook november 23,...
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![Page 1: S56 (5.1) The Circle.notebook November 23, 2016€¦ · S56 (5.1) The Circle.notebook November 23, 2016 Daily Practice 9.11.2016 Q1. Write 5x 2 - 10x + 8 in completed square form](https://reader030.vdocuments.us/reader030/viewer/2022041002/5ea3ce9ad40c105e7c400f80/html5/thumbnails/1.jpg)
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)
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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)
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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)
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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.
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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
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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
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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.
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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.
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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.