
Conic
Sections 


Ellipse

Definition and construction 
Eccentricity and linear
eccentricity

Constructions of an ellipse 
Equation of the ellipse, standard
equation of the ellipse

The equation of the ellipse examples 





Definition and construction, eccentricity and linear
eccentricity

An ellipse is the set of points (locus) in a plane whose distances from two fixed points have a constant sum. 
The fixed points F_{1}
and F_{2 }are called foci. Thus, the sum of distances
of any point P, of the ellipse, from the

foci is

F_{1}P 
+

F_{2}P  =
2a, and the
distance between foci 
F_{1}F_{2 } =
2c, 
where 2a is the
major
axis,
2c is called
focal distance and
c
is also called linear
eccentricity. 
Quantities
a
and c,
a
> c,
uniquely determine an ellipse. The ratio e = c/a,
e < 1
is called eccentricity
of the ellipse. 
The ellipse can also be defined as the locus of points the ratio of whose distances from the
focus to a vertical line, known as the
directrix ( d
), is a constant
e,
where 





Constructions of an ellipse 
According to
definition we
will explain two constructions of the ellipse: 


Fasten the ends of a string of length
2a
> 2c
at two distinct points F_{1} and
F_{2}. 
Pull the loop of string tight using the pencil until a triangle is formed with the pencil and the two foci as vertices. 
Keeping the string pulled tight, move the pencil around until the ellipse is traced out. 




 On a given line segment
A_{1}^{'
}A_{2}^{' }
= 2a, of the major
axis, choose an arbitrary point R.

Draw an arc of the radius A_{1}^{'}R
= r_{1}
with center F_{1}
and then draw an arc with center F_{2
}and radius A_{2}^{'}R
= r_{2}, intersecting the arc at points P_{1} and
P_{2}. 
Repeat the same procedure by drawing the arc of the radius
r_{1
}centered at F_{2
}and the arc of the radius
r_{2}
with
the center F_{1 }to obtain intersections
P_{3} and
P_{4}.

Using this method we can draw as many points of the ellipse as needed, noticing that while choosing point R, always must be
r_{1}
> a 
c and r_{2}
>
a 
c. 
This construction shows that the ellipse has two axes of symmetry of different length, the
major and
minor axes. Their intersection is the center of the ellipse. 




Equation of the ellipse, standard
equation of the ellipse

If in the direction of axes we introduce a coordinate system so that the center of the ellipse coincides with the 
origin, then coordinates of foci are 
F_{1}(c, 0) and
F_{2}(
c, 0). 
For every
point P(x,
y) of the ellipse, according to 
definition
r_{1} + r_{2} =
2a, it follows
that 

after
squaring 




and
reducing 


Repeated squaring and grouping gives 
(a^{2}

c^{2}) · x^{2}
+ a^{2}y^{2} = a^{2}
· (a^{2} 
c^{2}), 
and
since 
a^{2}

c^{2} = b^{2} 


follows 
b^{2}x^{2}
+ a^{2}y^{2} = a^{2}b^{2} 
equation
of the ellipse, 




and after division by
a^{2}b^{2}, 

standard equation of the ellipse. 


It follows from the equation that an ellipse is defined by values of
a
and b, or as they are associated through
the relation a^{2}

c^{2} = b^{2},
we can say that it is defined by any pair of these three quantities. 
Intersections of an ellipse and the coordinate axes we determine from equation 

by
putting, 

y
= 0 =>
x
= + a,
so obtained are vertices at the ends of the major axis A_{1}(a,
0) and
A_{2}(a,
0), and 
x
= 0 =>
y = + b,
obtained are covertices, the endpoints of the minor axis B_{1}(0,
b) and
B_{2}(0,
b). 
The line segments
A_{1}A_{2
}= 2a and
B_{1}B_{2
} = 2b are the
major and
minor axes while
a and b
are the
semimajor and
semiminor axes respectively. So the arc of the radius
a
centered at B_{1}and
B_{2 }intersects the major axis at
the foci F_{1}and
F_{2}. 
The focal parameter, called latus rectum and denoted
2p, is the chord perpendicular to the major axis
passing through any of the foci, as shows the above figure. The length of which equals the absolute value of ordinates of the points of the ellipse whose abscissas x
= c
or x
= c
that is 

so
the length of the latus rectum 

the
length of the semilatus rectum of the ellipse. 


The
equation of the ellipse examples 
Example:
Given is equation of the ellipse
9x^{2}
+ 25y^{2} = 225, find the lengths of semimajor and
semiminor axes, coordinates of the foci, the eccentricity and the length of the
semilatus rectum. 
Solution: From the standard equation 

we can find the semiaxes
lengths dividing the given 

equation by
225, 


coordinates of the
foci F_{1}(c, 0) and
F_{2}(
c, 0),
since




Example:
From given quantities of an ellipse determine remaining unknown quantities and write equation
of the ellipse, 

Solution:
a) Using 


therefore,
the semiminor axis 


the
linear eccentricity 


the
semi latus rectum 

and
the equation of the ellipse 





the eccentricity 

and
the equation of the ellipse 




the
semi latus rectum 

and
the equation of the ellipse 



d) unknown quantities expressed through given values, 


Example:
Find the equation of
the ellipse whose focus is F_{2}(6,
0) and which passes through the point
A(5Ö3,
4). 
Solution:
Coordinates of the point
A(5Ö3,
4) must satisfy equation of the ellipse, therefore 


thus,
the equation of the ellipse 




Example:
Write equation of the ellipse passing through points
A(4,
2) and B(8,
1). 
Solution:
Given points must satisfy equation of the ellipse, so 

Therefore,
the equation of the ellipse 

or
x^{2}
+ 16y^{2} = 80. 









Intermediate
algebra contents 



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