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is the set of all points in a plane,whose distances from a fixed point in the plane and their distances from a fixed line in the plane bear a constant ratio,less than 1. The fixed point is called focus
, the fixed line directrix
and the constant ratio 'e' < 1, the eccentricity of the ellipse.
The two fixed points are called the foci (plural of ‘focus’) of the ellipse.
The mid point of the line segment joining the foci is called the center of the ellipse. The line segment through the foci of the ellipse is called the major axis and the line segment through the center and perpendicular to the major axis is called the minor axis. The end points of the major axis are called the vertices of the ellipse.
The eccentricity of an ellipse is the ratio of the distances from the center of the ellipse to one of the foci and to one of the vertices of the ellipse (eccentricity is denoted by e) e = c / a
Standard equations of an ellipse The equation of an ellipse is simplest if the center of the ellipse is at the origin
and the foci are
if a > b
Observations on the basis of standard equation
If (x,y) satisfies the standard equation then (-x,-y),(x,-y) and (-x,y) also satisfy the standard equation of ellipse. This shows that the ellipse is symmetrical with respect to both the coordinate axes and the origin. Because of this the ellipse has two foci and two directrices.
Since b = a √(1-e2
), it shows that 0 < b < a.
If a < b
then the equation of an ellipse in simplest form is
Center is other than origin :
If the coordinates of center (h,k) then the equation of ellipse is
Here , center ( 2,3 ) and a= 3 and b = 2 so a > b .
∴ Ellipse main axis is X-axis.( Horizontal Ellipse)
Here, center ( -2, -3 ) and a = 2 and b = 4 so a< b
∴ Ellipse main axis is Y- axis. (Vertical Ellipse).
Russia-Ukraine crisis update - 3rd Mar 2022
The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.