
Conic
Sections 


Ellipse

Equation of a translated ellipse

Ellipse examples 





Equation of a translated 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. 

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:
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. 








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