Ellipse is a circle which is squashed into oval shape. In other words a circle is an ellipse where both foci are at the same end.
A curved line forming a closed loop, where the sum of the distances from the two foci to every point on the line is constant. or
Set of all points for which the sum of the distance from the two foci are always constant.
Important names used:
Derivation of standard equation:
Let S and S' be the foci and DD' be the directrix. Draw SZ perpendicular to A,A' dividing SZ internally and externally in the ratio e:1 where 'e' is the eccentricity. then A, A' are points on the curve.
Let C be the mid point of AA' and let AA' = 2a.
Take CA to be the X axis and CB is perpendicular to be the Y-axis. C is the origin.
SA/AZ = e, SA'/A'Z =e
SA = eAZ
A'S = e A'Z
SA + A'S = e(AZ +A'Z)
AA' = e(CZ-CA+A'C+CZ)
2a = e(2CZ)
CZ = a/e
A'S-SA = e(A'Z-AZ)
A'S+CS-(CA-CS) = e(AA')
2CS = e.2a
CS = ea
So, S is the point (ae,0)
Let P(x,y) be any point on the elipse. Drawing PM and PN perpendicular to DD' and CZ respectively.
PM= NZ = Cz-CN= (a/e)-x.
Since the point P lies on the curve
SP² = e²PM²
(x-ae)²+y² = e²(a/e-x)²
x²-2aex+y² = e²-2aex+e²x²
x²(1-e²)+y² = a²(1-e²)
x²/a² + y²/(a²(1-e²)) = 1.
If we put b² = a²(1-e²), then
x²/a² + y²/b² =1. This is the equation in standard form.
Parents and teachers can guide the students to learn how to derive the standard equation of ellipse from the above method. If you have any doubts please contact us, we will help you to clear your doubts.