s56 (5.3) the circle.notebook august 27, 2015 -...
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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