Ellipse

ELLIPSE It is the path traced by a point which moves in a plane in such a way that the sum of its distance from two fixed points in the plane is a constant.

The two fixed points are called the foci of the ellipse.

NOTE The plural of focus is foci.

In the given figure, F₁ and F₂ are two

fixed points and P is a point which moves

in such a way that PF₁ + PF₂ = constant.

The path traced by the point P is called an

ellipse, and the points F₁ and F₂ are called

its foci

SOME MORE TERMS RELATED TO AN ELLIPSE

(I) CENTRE OF THE ELLIPSE

The midpoint of the line segment joining the foci, is called the centre of the ellipse. In the given figure, F₁ and F₂ are the foci of the ellipse and O is its centre,

where OF₁ = OF₂

(II) AXES OF THE ELLIPSE MAJOR AXIS: The line segment through the foci of the ellipse with its end points onthe ellipse, is called its major axis.

In the given figure, AB is the major axis of the ellipse.

MINOR AXIS: The line segment through the centre and perpendicular to the major axiswith its end points on the ellipse, is called its minor axis.

In the given figure, CD is the minor axis of the ellipse.

(III) VERTICES OF AN ELLIPSE

The end points of the major axis of an ellipse are called its vertices.

In the given figure, A and B are the vertices of the ellipse.

AN IMPORTANT NOTE In an ellipse, we take:

Length of the major axis = AB = 2a .

Length of the minor axis = = CD = 2b .

Distance between the foci = = F₁₂ F₁₂ = 2c .

Length of the semi-major axis = a.

Length of the semi-minor axis = b.

(IV) ECCENTRICITY OF AN ELLIPSE

The ratio "c/a" is always constant and it is denoted by e, called the eccentricity of

the ellipse.

For an ellipse, we have 0 < e < 1

AN IMPORTANT RESULT

HORIZONTAL ELLIPSE

In the given equation of an ellipse, if the coefficient of x² has the larger denominator then its major axis lies along the x-axis

VERTICAL ELLIPSE

In the given equation of an ellipse, if the coefficient of Y² has the larger denominator then its major axis lies along the Y-axis

LATUS RECTUM OF A HORIZONTAL ELLIPSE

The latus rectum of an ellipse is a line segment perpendicular to the major axis,

passing through any of the foci with end points lying on the ellipse.