DERIVATION OF STANDARD EQUATION FOR ELLIPSE
FROM THE LOCUS DEFINITION OF AN ELLIPSE
An ellipse is the set of all points for which the sum of the distances from two fixed points
(foci) is constant:
( x − (−c))2 + ( y − 0) 2 + ( x − c) 2 + ( y − 0) 2 = constant
By looking at the case when the point on the ellipse is a vertex and adding up the
distances from that vertex to each focus, we know that the sum of the distances is 2a.
( x − (−c))2 + ( y − 0) 2 + ( x − c) 2 + ( y − 0) 2 = 2a
By looking at the case when the point on the ellipse is a covertex we find that the distance
2
2
2
from a focus to a covertex is a and use the Pythagorean theorem to find that a = b + c
The rest of the proof is algebra, rearranging and simplifying the distance formula
to eliminate the square roots and obtain the standard equation of the ellipse
( x − ( − c )) 2 + ( y − 0 ) 2 + ( x − c ) 2 + ( y − 0 ) 2 = 2 a
( x + c) 2 + y 2 + ( x − c) 2 + y 2 = 2a
( x + c) 2 + y 2 = 2a − ( x − c) 2 + y 2
(
( x + c) 2 + y 2 = 2a − ( x − c) 2 + y 2
(moved one radical to the other side)
)
2
(squared both sides)
x 2 + 2cx + c 2 + y 2 = 4a 2 − 4a ( x − c ) 2 + y 2 + ( x − c ) 2 + y 2
x 2 + 2cx + c 2 + y 2 = 4a 2 − 4a ( x − c ) 2 + y 2 + ( x − c ) 2 + y 2
x 2 + 2cx + c 2 + y 2 = 4a 2 − 4a ( x − c) 2 + y 2 + ( x − c) 2 + y 2
x 2 + 2cx + c 2 + y 2 = 4a 2 − 4a ( x − c) 2 + y 2 + x 2 − 2cx + c 2 + y 2
2cx = 4a 2 − 4a ( x − c) 2 + y 2 − 2cx
4cx − 4a 2 = 4a ( x − c) 2 + y 2
cx − a 2 = a ( x − c) 2 + y 2
(cx − a )
2 2
(
= a 2 ( x − c) 2 + y 2
(finally finished simplifying!!!)
)
(squared both sides, again)
c 2 x 2 − 2a 2 c 2 x 2 + a 4 = a 2 x 2 − 2a 2 c 2 x 2 + a 2 c 2 + a 2 y 2
c 2 x 2 + a 4 = a 2 x 2 + a 2c 2 + a 2 y 2
a 4 − a 2c 2 = a 2 x 2 − c 2 x 2 + a 2 y 2
(
) (
)
a2 a2 − c2 = a2 − c2 x2 + a2 y2
( finished simplifying, again)
but a2 = b2 + c2 so a2 −c2 = b2
a 2b 2 = b 2 x 2 + a 2 y 2
a 2b 2 b 2 x 2 a 2 y 2
=
+
a 2b 2 a 2 b 2 a 2b 2
x2 y2
Finally we simplify to 1 = 2 + 2
a
b
(substituted b2 )
(divide both sides to get 1 on the constant side )
, standard form of equation of an ellipse