MATHEMATICS EXTENSION 2
4 UNIT MATHEMATICS
TOPIC 3: CONICS
3.1
THE ELLIPSE
Ellipses are the closed type of conic sections: a plane curve that results
from the intersection of a cone by a plane. Ellipses have many similarities
with the other two forms of conic sections: the parabolas and the
hyperbolas, both of which are open and unbounded. The cross section of
a cylinder is an ellipse, unless the section is parallel to the axis of the
cylinder.
An ellipse obtained as the intersection of a cone with an inclined plane.
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An ellipse can be defined as the locus of all points that satisfy the equation
( x  x1 )2 ( y  y1 )2

1
a2
b2
Variables:
(x, y) the coordinates of any point on the ellipse
Constants: (x1, x2) the coordinates of the ellipse's centre
a, b are the radius on the X and Y axes respectively
The circle is simply an ellipse where a = b. The equation of a circle of radius a and centre
(x1, x2) is
( x  x1 )2  ( y  y1 )2  a 2
The equations for an ellipse and circle centred on the origin (0, 0) are
x2 y2

1
a 2 b2
ellipse
x2  y 2  a2
circle
For the ellipse, a is best refer to as the X radius and b as the Y radius. When a > b, a is
often called the semi-major axis and b is called the semi-minor axis.
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As seen from the equation and graph of an ellipse it is a multi-valued function. For each
value of x there are two y values.
The equation for an ellipse with can also be
given in parametric form
x  x1  a cos  
y  y1  b sin  
where  is an angle which ranges from 0 to 2
radians.
The equation of a circle in parametric form with
the centre at the origin (0,0) and radius a is
x  a cos  
y  a sin  
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x  a cos   cos2    x 2 / a 2
y  b sin   sin 2    y 2 / b2
sin 2    cos2    x 2 / a 2  y 2 / b2  1
Do the online simulation activity in which you can view the shape an ellipse by changing the
values of a and b. Predict the changes in the shape of the ellipse. Then test your predictions
and account for any discrepancies between your predictions and observations.
View the online SIMULATION ACTIVITY
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each
planet in the solar system is an ellipse with Sun at one of the focal points. The same is true for
moons orbiting planets and all other systems having two astronomical bodies.
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An ellipse may also be defined as the locus of points, the sum of whose distances from two
fixed points F1 and F2 (the focal points or foci) is constant and equal to 2a.
For any point P on the ellipse:
F1P + F2P = 2 a
The circle is a special type of an ellipse that has both focal points at the same location. The shape
of an ellipse (how 'elongated' it is) is represented by its eccentricity e, which for an ellipse can be
any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.
semi-major axis a
semi-minor axis b
c
a 2  b2
b
e 
 1  
a
a
a
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focal length c
eccentricity e
2
 0  e 1
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An ellipse can also be defined as the set of points such that the ratio of the distance of each
point on the curve from a given point (called a focus or focal point) to the distance from that
same point on the curve to a given line (called the directrix) is a constant, called the eccentricity
e of the ellipse.
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Derivation of the Cartesian form of an ellipse from the focus-directrix definition
d1 / d 2  e d12  e 2 d 2 2
d12  ( x  c ) 2  y 2
d 2 2  ( x  a / e) 2
c  e a e  c / a c 2  a 2  b2
a 2  b2
e 
1  e2  b2 / a 2
2
a
( x  e a ) 2  y 2  e 2 ( x  a / e) 2
2



y 2  x 2 e 2  1  x  2a e  2a e   a 2 1  e 2



y 2   x 2 b2 / a 2  b2
x2 y2

1
a 2 b2
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Derivation of the Cartesian form for an ellipse from the locus of points, the sum of whose
distances from two the two focal points is constant and equal to 2a.
You solve the problem by calculating the
distances d1 and d2 as in the above
example. However, the algebra becomes
“very messy”. For interest only, you can
view the full derivation on wikipedia.
https://en.wikipedia.org/wiki/Derivation_of_the_Cartesian_form_for_an_ellipse
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Equation of tangents to an ellipse
The equation of an ellipse with its centre at the origin is
x2 y2

1
a 2 b2
The equation of the tangent to the ellipse can be expressed as
y  M1 x  B1
where M1 is the gradient and B1 is the intercept of the straight line.
The first derivative dy / dx at a point P  xP , yP  on the ellipse gives the gradient of the tangent to
the ellipse at that point. The implicit differentiation of the equation for the ellipse gives
2 x 2 y dy

0 
a 2 b2 dx
2
dy
b x
  
dx
a y
The gradient M1 of the tangent at the point P  xP , yP  is
2
b x
M1     P
 a  yP
and the intercept B1 is
2
 b   xP 
B1  yP    

 a   yP 
2
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The equation of the tangent is
y  M 1 x  B1
2
b x
M1     P
 a  yP
2
 b   xP 
B1  y P    

 a   yP 
2
2
2
 b   xP 
 b   xP 
y    
 x  yP    

 a   yP 
 a   yP 
2
2
 y P 2   xP 2 
 yP 
 xP 
 2  y   2  x   2    2 
b 
a 
 b  a 
 xP 2   y P 2 
 2   2  1
a   b 
 xP 
 yP 
x

 2
 2  y 1
a 
b 
The tangent crosses the X-axis (y = 0) at the point T  xT ,0  where
a2
xT 
xP
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Equation of a normal to an ellipse
The normal to an ellipse at the point P  xP , yP  is a straight line that is perpendicular to the
tangent at that point.
Equation of tangent
y  M 1 x  B1
Equation of normal
y  M 2 x  B2
Gradient of the normal M2
1  a  y P
M2 
 
M 1  b  xP
2
Intercept of normal B2
 b2  a 2 
a
B2  yP    yP  yP 

2
b
 b

2
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The equation of the normal is
y  M 2 x  B2
2
a y
M2    P
 b  xP
 b2  a 2 
B2  y P 

2
 b

2
a y
y    P x  yP
 b  xP
 b2  a 2 


2
 b

 a2 
 b2 
2
2
  x  y  a b
 xP 
 yP 
The normal crosses the X-axis (y = 0) at the point N  xN ,0  where
 a 2  b2 
xN  
 xP
2
 a

No is no need to remember the formulae for the tangent and normal to the ellipse. It is best to
derive the equations from first principles.
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Examples
Consider the ellipse 9 x 2  25 y 2  225  0
Verify all the numerical values and information in the following two figures for two points on the
ellipse at x = 3.5 and x = -3.8.
Note: the reflection property of an ellipse – the normal to an ellipse at a point P on it is equally
inclined to the focal chords through P 1   2
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