###### In the previous article , we solved few solutions of Short Answer Type Questions of Ellipse Chapter of S.N.Dey mathematics, Class 11. In this chapter, we will solve few long answer type questions.

1. Find the centre, vertices, equations of the axes, lengths of the axes, eccentricity, the length of latus rectum, co-ordinates of foci and the equations of the directrices of each of the following ellipses :

Solution (a).

We have the given equation of ellipse

Comparing with the general equation of ellipse we get,

So, the major axis of the ellipse is parallel to the axis and minor axis is parallel to -axis.

(i) Centre

(ii) Vertex i.e. and

(iii) Major axis is given by and minor axis is given by

(iv) Length of major axis unit and length of minor axis unit.

(v) Eccentricity

(vi) Length of latus rectum unit.

(vii) The co-ordinates of foci are given by

i.e. and

(viii) The equation of the directrix is

Solution (b).

We have the given equation of ellipse

Comparing with the general equation of ellipse we get,

Since the major axis of the ellipse is parallel to the axis and minor axis is parallel to -axis.

(i) Centre

(ii) Vertex i.e. and

(iii) Major axis is given by and minor axis is given by

(iv) Length of major axis unit and length of minor axis unit.

(v) Eccentricity

(vi) Length of latus rectum unit.

(vii) The co-ordinates of foci are given by

i.e. and

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Solution (c).

We have the given equation of ellipse

Comparing with the general equation of ellipse we get,

Clearly, the major axis of the ellipse is parallel to the axis and minor axis is parallel to -axis.

(i) Centre

(ii) Vertex i.e. and

(iii) Minor axis is given by and major axis is given by

(iv) Length of major axis unit and length of minor axis unit.

(v) Eccentricity

(vi) Length of latus rectum unit.

(vii)The co-ordinates of foci are given by

i.e. and

(viii) The equation of the directrix is

(viii) The equation of the directrix is

Find the eccentricity, the lengths of latus rectum and the centre of the ellipse

Solution.

We have the given equation of ellipse

From we can say that the major axis of the given ellipse is parallel to axis and minor axis of the ellipse is parallel to axis.

The eccentricity of the ellipse is

The length of the latus rectum is

Finally, the centre of the ellipse is

Find the latus rectum, the eccentricity and the co-ordinates of the foci of the ellipse

Solution.

The equation of the ellipse

Comparing with the general equation of ellipse we get,

The length of the latus rectum

.

The eccentricity of the ellipse is

The co-ordinates of foci are

i.e., and

i.e., and .

Examine, with reasons, the validity of the following statement :

represents the equation of an ellipse whose eccentricity is

Solution.

Comparing with the general equation of ellipse we get, .

The eccentricity of is

So, the given equation represents the equation of the ellipse with eccentricity

Hence, the aforesaid statement is valid.