Conic Section, Standard Equation an Ellipse, Major Axis, Minor Axis

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Standard Equation an Ellipse

Let be the focus, be the Directrix and be a moving point. Draw perpendicular from on the Directrix. Let be the eccentricity.

Divide internally and externally at and (on produced) respectively in the ratio 1, as .

And

Since and are points such that their distances from the focus bears a constant ratio to their respective distances from the Directrix and so they lie on the ellipse. These points are called vertices of the ellipse.

Standard Equation an Ellipse

Standard Equation an Ellipse

Standard Equation an Ellipse

Let be equal to and be its mid point, i.e.,

The point is called the centre of the ellipse.

Adding (1) and (2), we have

Or

Or

Or

Subtracting (1) from (2), we have

Or

Or

Or

Let us choose as origin, as x-axis and , a line perpendicular to as y-axis.

Coordinates of are then and equation of the Directrix is

Let the coordinates of the moving point be . Join P, draw .

By definition

Or

Or

Or

Or

Or

Or [On dividing by ]

Putting , we have the standard form of the ellipse as .

Major axis: The line joining the two vertices and , i.e., is called the major axis and its length is .

Minor axis: The line passing through the centre perpendicular to the major axis, i.e., is called the minor axis and its length is .

Principal axis: The two axes together (major and minor) are called the principal axes of the ellipse.

Latus rectum: The length of the line segment is called the latus rectum and it is given by

Equation of the Directrix:

Eccentricity: is given by

Example:

Find the equation of the ellipse whose focus is , eccentricity and the Directrix is .

Solution:

Let be any point on the ellipse then by the definition, its distance from the focus. Its distance from Directrix or

( is the foot of the perpendicular drawn from to the directrix).

Or

Or

The locus of is,

Which is the required equation of the ellipse.

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