

Ellipse

The parametric equations of the ellipse 
Equation of a translated ellipse

Equations 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. 

Equations
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. 


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. 








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