###### In this article, we will do few solutions of Short Answer Type Questions of Ellipse Chapter of S.N.Dey mathematics, Class 11.

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

Solution (a).

The given equation of the ellipse can be written as

Comparing with the general form of the ellipse we get,

Clearly, major axis of the ellipse is along axis , minor axis of the given ellipse is along axis and the centre of the ellipse is at

(i) The length of the major axis is and

the length of the minor axis is

(ii) The length of the latus rectum is

(iii) The co-ordinates of vertices

(iv) Eccentricity of the ellipse is given by

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

(vi) Equations of directrices are given by

Solution(b)

The given equation of the ellipse can be written as

Comparing with the general form of the ellipse we get,

Clearly, major axis of the ellipse is along axis , minor axis of the given ellipse is along axis and the centre of the ellipse is at

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(i) The length of the major axis is and

the length of the minor axis is

(ii) The length of the latus rectum is

(iii) The co-ordinates of vertices

(iv) Eccentricity of the ellipse is given by

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

(vi) Equations of directrices are given by

Solution(c)

The given equation of the ellipse can be written as

Comparing with the general form of the ellipse we get,

Clearly, major axis of the ellipse is along axis , minor axis of the given ellipse is along axis and the centre of the ellipse is at

(i) The length of the major axis is and

the length of the minor axis is

(ii) The length of the latus rectum is

(iii) The co-ordinates of vertices

(iv) Eccentricity of the ellipse is given by

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

(vi) Equations of directrices are given by

Solution(d)

The given equation of the ellipse can be written as

Comparing with the general form of the ellipse we get,

(i) The length of the major axis is and

the length of the minor axis is

(ii) The length of the latus rectum is

(iii) The co-ordinates of vertices

(iv) Eccentricity of the ellipse is given by

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

(vi) Equations of directrices are given by

Find the eccentricity and equations of the directrices of the ellipse Show that the sum of the focal distances of any point on this ellipse is a constant.

Solution.

Comparing the given equation of ellipse with the general form of the ellipse we get,

Eccentricity of the ellipse is given by

Equation of the directrix is given by

Any point on the given ellipse can be written as

The co-ordinates of the foci are given by

Now, the sum of the focal distances of the point P is

Note[*] :

Taking major and minor axes as and -axes respectively, find the equation of the ellipse

whose length of major and minor axes are and respectively.

Solution.

By question, the equation of the ellipse can be written as

(i) The length of the major axis

and the length of the minor axis

Hence, by the equation of the ellipse is

whose lengths of minor axis and latus rectum are and

Solution.

Length of the minor axis

Length of the latus rectum

Hence, the equation of the ellipse is given by

whose eccentricity is and co-ordinates of foci are

Solution.

By question, the equation of the ellipse can be written as

The eccentricity

The co-ordinates of foci

So, the equation of the ellipse is given by

whose eccentricity is and length of latus rectum

Solution.

Note[*] :

Again,

Hence, the equation of the ellipse is

which passes through the points and

Solution.

The equation of the ellipse can be written as

Since the ellipse passes through the points and

From and we get

Similarly, from and we get

Hence, replacing the values of and , we get from

Finally, from we get the required equation of ellipse whose major axis is along axis and minor axis is along -axis.

whose eccentricity is and passes through the point

Solution.

By question,

Since the ellipse passes through the point so

Hence, using and we get the required equation of ellipse which is

whose co-ordinates of vertices are and the co-ordinates of the ends of minor axes are

Solution.

By question, the co-ordinates of the vertices the co-ordinates of the end of minor axes

Hence, the equation of the ellipse is

whose distance between the foci is and the distance between the directrices is

Solution.

By question, the distance between the foci is and the distance between the directrices is

From and we get,

Again, from we get,

So, the equation of the ellipse is given by

whose eccentricity is and the sum of squares of major and minor axes is

Solution.

By question,

From and we get,

Hence, the equation of the ellipse is given by

whose co-ordinates of vertices are and the co-ordinates of one focus are

Solution.

By question, the equation of the ellipse can be taken in the form

The co-ordinates of the vertices i.e.

The co-ordinates of one focus

Hence, by we get the required equation of ellipse

whose length of latus rectum is unit and the co-ordinates of one focus are

Solution.

The length of latus rectum

Co-ordinates of one focus

So,

Hence, the equation of the ellipse is

(xii) whose distance between the foci is unit and minor axis is of length 4 unit.

Solution.

The length of the minor axis

Distance between the foci

Hence, the equation of the ellipse is

whose eccentricity is and the length of semi-latus rectum is unit.

Solution.

Eccentricity

The length of semi-latus rectum

From and we get,

So,

Hence, the equation of the ellipse is