The square and the rectangle shown below are equivalent figures. Each side of the square measures (x)
cm. The area of the rectangle is (2x2  7x  30) cm2.
(x) cm
Area: (2x2  7x  30) cm2
What is the perimeter of the rectangle? Your final answer must be a number.
Example of an appropriate method
Value of x
Since the square and the rectangle are equivalent, their areas are equal.
Area of the rectangle = Area of the square
2x2  7x  30 = x2
x2  7x  30 = 0
x2  10x + 3x  30 = 0
x(x  10) + 3(x  10) = 0
(x  10)(x + 3) = 0
x = 10 or
x = -3 (impossible)
Algebraic expressions representing the dimensions of the rectangle
2x2  7x  30
2x2  12x + 5x  30
2x(x  6) + 5(x  6)
(x  6)(2x + 5)
The rectangle measures (x  6) cm by (2x + 5) cm.
Perimeter of the rectangle
Since x = 10, the rectangle measures 4 cm by 25 cm.
Perimeter: 2(4 + 25) = 58 cm
Answer:
The perimeter of the rectangle is 58 cm.
The cylinders below are similar solids.
V=?
V = 72 cm3
Ab = 144 cm2
Ab = 16 cm2
The area of the base of the smaller cylinder is 16 cm2, and its volume is 72 cm3.
The area of the base of the larger cylinder is 144 cm2.
What is the volume of the larger cylinder to the nearest cubic centimetre?
Ratio of areas of bases =
 Scale factor =
144 cm 2 9
=
1
16 cm 2
9 3
=
1 1
3
 3
Ratio of volumes =  
1
27
=
1
Volume larger
27

=
3
1
72 cm
Volumelarger = 27  72 cm3
= 1944 cm3
Given the circle with the equation x2 + y2 + 4x  8y  5 = 0.
Which of the following equations defines the tangent to this circle through point P(2, 1)?
A)
3x  4y  5 = 0
C)
4x + 3y  11 = 0
B)
4x  3y  5 = 0
D)
4x  3y  11 = 0
Equation of the large circle in standard form
x2 + y2 + 4x  8y  5= 0
(x2 + 4x + 4) + (y2  8y + 16) = 5 + 4 + 16
(x + 2)2 + (y  4)2 = 25
(x + 2)2 + (y  4)2 = 52
Centre of circle : (-2, 4)
Slope of radius
m
y 2  y1
1 4
3


x2  x1  2  (2)
4
Slope of tangent :
4
3
Equation of tangent
y  ax  b
4
xb
3
4
1  ( 2)  b
3
5
b 
3
4
5
y  x
3
3
4
5
3( y  x  )
3
3
3y  4x  5
0  4x  3y  5
y
Result : The answer is B
The major axis of an elliptically-shaped lake measures 400 m while its minor axis measures 200 m. A
buoy is placed at each focus of the ellipse thus formed. What is the distance between the buoys?
The distance between the buoys is 346.4 m.
Note : a = 200, b = 100 and c = 173.2
y
An American spacecraft completed three circular
orbits around the Earth before continuing on its
journey towards the moon following a trajectory
which is tangent to its previous path.
B(12, 6)
A(8, 4)
x
In the adjacent diagram, the Earth's center is represented by point A.
Point B is the spacecraft's position at the moment it began its journey towards the moon. What is the
equation of the spacecraft's trajectory towards the moon?
The equation of the spacecraft's trajectory is y = -2x + 30
In the adjacent diagram, the centre of the circle is
located at the upper focus of the ellipse defined
by :
y
F
x2
y2

1
9
36
A
0
B
The circle passes through the ends, A and B, of the
minor axis of the ellipse.
What is the degree measure of arc AB, situated inside the ellipse?
Equation of ellipse
x2
y2

1
9
36
Given this equation
a = 9 =3
(half-length of the minor axis of the ellipse)
b = 36 = 6
(half-length of the major axis of the ellipse)
Measure of side FB = 6 units (half-length of major axis of the ellipse)
Since triangle FBO is a right triangle, then
sin  OFB =
m OB 3 1
= = and m  OFB = 30
m FB 6 2
Measure of angle AFB = 30  2 = 60 (y-axis being the axis of symmetry of AFB)
The measure of arc AB is 60 since angle AFB is at the centre.
Result : The measure of arc AB is 60 degrees.
x
A town has just opened a new horse-racing track. The track, which is elliptical in shape, is
illustrated below. The axes of the ellipse measure 200 m and 60 m.
A
C
60 m
B
D
200 m
Two fences ( AB and CD ) are located 20 m from the ends of the major axis and are perpendicular to
this axis. What is the length of each of these fences?
The length of each fence is 36 m.
The line that passes through points A and B is tangent to the circle with centre C as illustrated to the
right.
The circle is defined by the equation
x2 + y2 + 8x  10y  59 = 0.
The coordinates of the point of tangency, A, are (2, 13).
What is the equation of the tangent?
The equation of the tangent is 3x + 4y  58 = 0
A(2, 13)
C
B
A hiking trail in Gatineau Park is elliptical in shape. Over time, walkers have made a path
between Rest Area A and Rest Area B.
The path is 200 metres from vertex C.
y
A
200 m
Footpath
400 m
C
x
B
1000 m
How many metres long is path AB?
The path is 320m long.
A dome, in the shape of a semi-ellipse, protects a tennis court, as shown below.
8m
?
20 m
3m
The height of the dome at the centre is 8 m and its span is 20 m. Cameras must be fixed to the roof of
the dome at a horizontal distance of 3 meters from its edges.
At what height are the cameras from the ground? (Round your answer off to the nearest hundredth.)
Example of an appropriate method
a = 10
b=8
Find the equation of the semi-ellipse centre (0, 0)
2
2
x
y

 1
100
64
Find the value of y at (7, y)
49
y2
= 1

100
64
49
y2
= 1 
100
64
y2 = 32.64
y  +5.71 (-5.71 is rejected)
Answer:
The cameras are 5.71 m from the ground.
(Students may use other centres.)