The document discusses key properties of ellipses: - An ellipse can be defined as all points where the sum of distances to two fixed foci is a constant. - The standard equation of an ellipse centered at the origin is x^2/a^2 + y^2/b^2 = 1, where a and b are the semi-major and semi-minor axes. - The location of the foci depend on the relative lengths of the axes, with the foci located a distance of c = √(a^2 - b^2) units from the center along the major axis. - The equation can be transformed to (x-h)^2/a^2 +

1. Conics:
The Ellipse

2. Definition
An ellipse can be defined as the locus
of points, or path traced out, in a
plane such that the sum of the
distances from the moving point to
two fixed points remains constant.
The two fixed points are called foci.

3. Equation
x2 + y2 = 1
a2 b2
a is ½ the horizontal axis
b is ½ the vertical axis

4. Different Ellipses
Vertical ellipse:
If b is greater than a
the ellipse is vertical.
The centre can
be translated to
(h,k)
Foci are on major
or long axis.

5. Focus
The focus determines the shape of
the ellipse.
The focus is represented by ‘c’.
c is the distance from the centre to
the foci along the major axis.
c2 = a2 - b2 where a>b
Or c2 = b2 – a2 where b>a
Where are the foci on the last page?

6. Solve for c
If a is 1 and b is 2, what do we do?
Do we use c2 = a2 - b2 or b2 – a2
So c2 = b2 – a2
c2 = 22 – 12
c2 = 4 – 1
c2 = 3
c = √3 = 1.736
So the foci are at
(0,1.736) and (0,-1.736)

7. Transformed Equation
(x-h)2 + (y-k)2 = 1
a2 b2
Where the centre is (h,k)
a is ½ the horizontal axis
b is ½ the vertical axis
Endpoints (h+a, k+b) (h-a, k-b)
(h+a, k-b) (h-a, k+b)
The centre is at the midpoint of the 2 axes.

8. Off Centre Ellipse
I keep the
x/a
y/b

9. Uses
Bluetooth Washington DC Ellipse

10. Elliptical Orbit

11. Exam Question
D) should be x2 + y2 = 9
What is the equation of the ellipse whose vertices are :
A1(-3, 0) A2(3, 0) B1(0, -3) B2(0, 3)
A) 1
33
22

yx
C) x2
+ y2
= 1
B) 1
99
22

yx
D) 2 + y2
= 9

12. Exam Question
A railway tunnel goes through a mountain. A cross section shows that the tunnel is in the shape
of the upper half of an ellipse whose equation is
9x2
+ 4y2
= 144
where the unit of measure is the metre.
The tunnel's lighting consists of lamps located at
the upper focus of the ellipse.
Rounded to the nearest hundredth, how far from
the ground are the lights?
A) 3.61 m C) 5.33 m
B) 4.47 m D) 7.21 m

13. Exam QuestionA 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
B
C
D
200 m
60 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?
A) 9 m C) 36 m
B) 24 m D) 59 m

14. Activity
Page 346, Q 1,3,4,6,7,10a,12