s56 (5.1) the circle.notebook november 23, 2016€¦ · s56 (5.1) the circle.notebook november 23,...
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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.