
Conic
Sections 


Ellipse

The parametric equations of the ellipse 
Equation of a translated ellipse

The equation of the ellipse examples 





The parametric equations of the
ellipse 
Equation of the ellipse in the explicit form 

can help us to explain another construction 

of
the ellipse. So, in the coordinate system draw two concentric circles of radii equal to lengths of the semi axes a and b, with the center at the origin as shows the figure. 
An arbitrary chosen line through the origin intersects the
circle of the radius a
at the point R
and the circle of radius b
at M.

Then, the parallel line with the major axis through M
intersects the parallel line with the minor axis through
R, at
a point P(x,
y) of the ellipse.
Proof, 
in
the figure, OS
= x,
PS
= y
and 


as the triangles
OMN
and ORS
are similar, then

OM
: OR
= MN
: RS
or b
: a
= PS
: RS,

so
that 





It proves that the point
P(x,
y) obtained by the construction
lies on the ellipse. This way, using the figure, we also derive 
the
parametric equations of
the ellipse 

where the parameter
t
is an angle 0
<
t
<
2p. 

By dividing the first parametric equation by
a
and the second by b, then square and add them, obtained is
standard equation of the ellipse. 

Equation of a translated ellipse
the ellipse with the center at
(x_{0},
y_{0})
and the major axis parallel to the xaxis. 
The equation of an ellipse that is translated from its standard position can be obtained by replacing
x
by x_{0} 
and
y by
y_{0} in its standard equation, 



The above equation can be rewritten into
Ax^{2}
+ By^{2} + Cx + Dy + E = 0. 
Every equation of that form represents
an ellipse if A
not equal
B
and A ·
B > 0 that is, if the square terms have unequal coefficients, but the same
signs. 

The
equation of the ellipse examples 
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. 


Example:
In the ellipse 4x^{2}
+ 9y^{2} = 144 inscribed is a rectangle whose vertices lie on the ellipse and
whose sides are parallel with ellipse axes. Longer side, which is parallel to the major axis, relates to the shorter
side as 3 :
2. Find the area of the rectangle. 
Solution: It follows from the given condition that the coordinates
of vertices of the rectangle must satisfy the 
same ratio, i.e.,
x :
y
= 3 :
2
=>
x =
3y/2.

To determine points of the ellipse of which coordinates are in this ratio, put these variable
coordinates into equation of the ellipse, 
P(3y/2,
y) =>
4x^{2}
+ 9y^{2} = 144 
4(3y/2)^{2}
+ 9y^{2} = 144 =>
18y^{2} = 144,

y_{1,2} =
±Ö144/18 =
±2Ö2,
x =
3y/2
=>
x_{1,2} =
±3Ö2.

Therefore, the vertices of the rectangle,




A(3Ö2,
2Ö2),
B(3Ö2,
2Ö2),
C(3Ö2,
2Ö2)
and D(3Ö2,
2Ö2). 
The
area of the rectangle A =
4 ·
x
·
y =
4 ·
(3Ö2
)
· (2Ö2
)
=
48 square units. 

Example:
Given is equation of the ellipse
4x^{2}
+ 9y^{2}
+ 24x 18y
+ 9 =
0,
find its center S(x_{0},
y_{0}),
the semiaxes and intersections of the ellipse with the coordinate axes. 
Solution: Coordinates of the center and the semiaxes are shown
in the equation of the translated ellipse, 

Rewrite the given equation to that
form, 
4(x^{2}
+ 6x)
+ 9(y^{2} 
2y)
+ 9 = 0 
4[(x
+ 3)^{2} 
9] +
9[(y 1)^{2}
1]
+ 9 = 0

4(x + 3)^{2}
+ 9(y 1)^{2}
= 36
or





therefore,
S(3,
1),
a = 3 and
b = 2.

Intersections of the ellipse and the
xaxis
we obtain by setting y =
0 into the equation of the ellipse, thus

4x^{2}
+ 24x + 9 =
0,
x_{1,2} = 3
± 3Ö3/2, 
and intersections of the ellipse
with the yaxis by
setting x
= 0, =>
9y^{2}
18y
+ 9 =
0,
y_{1,2} = 1. 








Intermediate
algebra contents 



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