## Ellipse

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. Definition:

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.

•  Here e < 1.

Important names used:

• Center: A mid point in the line segment linking the two foci of the curve. In other words intersection of major and minor axes.
• Major and minor axes: The longest and the shortest diameter of the curve. Both axes are perpendicular bisectors of each other. If both the axes are equal in length, then this curve becomes a circle. The foci always lie in the major axis.
• Foci(two focus points): Ellipse is defined by foci. Foci always lie on the major axis.
• Tangent: A line which touches the curve at one point.
• Chord: A line which joins two point on the curve.

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

SP/PM =e

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.

1. In the above equation if we put y=0, x = ±a. The curve cuts the X-axis at A(a,0) and A'(-a,0)
2. On the other hand if we put x=0 and y= ±b, the curve cuts Y-axis at the points (0,b) and B'(0,-b)
3. AA' =2a is called the major axis BB' =2b is called the minor axis and            a>b.

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