## NCERT Solutions for Class 11 Maths Chapter 11 – Conic Sections Ex 11.3

#### Page No 255:

#### Question 1:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*x*-axis, while the minor axis is along the*y*-axis.
On comparing the given equation with, we obtain

*a*= 6 and*b*= 4.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (6, 0) and (–6, 0).

Length of major axis = 2

*a*= 12
Length of minor axis = 2

*b*= 8
Length of latus rectum

#### Question 2:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*y*-axis, while the minor axis is along the*x*-axis.
On comparing the given equation with, we obtain

*b*= 2 and*a*= 5.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, 5) and (0, –5)

Length of major axis = 2

*a*= 10
Length of minor axis = 2

*b*= 4
Length of latus rectum

#### Question 3:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*x*-axis, while the minor axis is along the*y*-axis.
On comparing the given equation with, we obtain

*a*= 4 and*b*= 3.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are.

Length of major axis = 2

*a*= 8
Length of minor axis = 2

*b*= 6
Length of latus rectum

#### Question 4:

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*y*-axis, while the minor axis is along the*x*-axis.
On comparing the given equation with, we obtain

*b*= 5 and*a*= 10.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±10).

Length of major axis = 2

*a*= 20
Length of minor axis = 2

*b*= 10
Length of latus rectum

#### Question 5:

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*x*-axis, while the minor axis is along the*y*-axis.
On comparing the given equation with, we obtain

*a*= 7 and*b*= 6.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (± 7, 0).

Length of major axis = 2

*a*= 14
Length of minor axis = 2

*b*= 12
Length of latus rectum

#### Question 6:

#### Answer:

The given equation is.

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*y*-axis, while the minor axis is along the*x*-axis.
On comparing the given equation with, we obtain

*b*= 10 and*a*= 20.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±20)

Length of major axis = 2

*a*= 40
Length of minor axis = 2

*b*= 20
Length of latus rectum

#### Question 7:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 36

*x*^{2}+ 4*y*^{2}= 144#### Answer:

The given equation is 36

*x*^{2}+ 4*y*^{2}= 144.
It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*y*-axis, while the minor axis is along the*x*-axis.
On comparing equation (1) with, we obtain

*b*= 2 and*a*= 6.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±6).

Length of major axis = 2

*a*= 12
Length of minor axis = 2

*b*= 4
Length of latus rectum

#### Question 8:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 16

*x*^{2}+*y*^{2}= 16#### Answer:

The given equation is 16

*x*^{2}+*y*^{2}= 16.
It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*y*-axis, while the minor axis is along the*x*-axis.
On comparing equation (1) with, we obtain

*b*= 1 and*a*= 4.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (0, ±4).

Length of major axis = 2

*a*= 8
Length of minor axis = 2

*b*= 2
Length of latus rectum

#### Question 9:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 4

*x*^{2}+ 9*y*^{2}= 36#### Answer:

The given equation is 4

*x*^{2}+ 9*y*^{2}= 36.
It can be written as

Here, the denominator of is greater than the denominator of.

Therefore, the major axis is along the

*x*-axis, while the minor axis is along the*y*-axis.
On comparing the given equation with, we obtain

*a*= 3 and*b*= 2.
Therefore,

The coordinates of the foci are.

The coordinates of the vertices are (±3, 0).

Length of major axis = 2

*a*= 6
Length of minor axis = 2

*b*= 4
Length of latus rectum

#### Question 10:

Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0)

#### Answer:

Vertices (±5, 0), foci (±4, 0)

Here, the vertices are on the

*x*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly,

*a*= 5 and*c*= 4.
It is known that.

Thus, the equation of the ellipse is.

#### Question 11:

Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)

#### Answer:

Vertices (0, ±13), foci (0, ±5)

Here, the vertices are on the

*y*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly,

*a*= 13 and*c*= 5.
It is known that.

Thus, the equation of the ellipse is.

#### Question 12:

Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0)

#### Answer:

Vertices (±6, 0), foci (±4, 0)

Here, the vertices are on the

*x*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly,

*a*= 6,*c*= 4.
It is known that.

Thus, the equation of the ellipse is.

#### Question 13:

Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (±3, 0), ends of minor axis (0, ±2)

#### Answer:

Ends of major axis (±3, 0), ends of minor axis (0, ±2)

Here, the major axis is along the

*x*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly,

*a*= 3 and*b*= 2.
Thus, the equation of the ellipse is.

#### Question 14:

Find the equation for the ellipse that satisfies the given conditions: Ends of major axis, ends of minor axis (±1, 0)

#### Answer:

Ends of major axis, ends of minor axis (±1, 0)

Here, the major axis is along the

*y*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly,

*a*= and*b*= 1.
Thus, the equation of the ellipse is.

#### Question 15:

Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5, 0)

#### Answer:

Length of major axis = 26; foci = (±5, 0).

Since the foci are on the

*x*-axis, the major axis is along the*x*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly, 2

*a*= 26 ⇒*a*= 13 and*c*= 5.
It is known that.

Thus, the equation of the ellipse is.

#### Question 16:

Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6)

#### Answer:

Length of minor axis = 16; foci = (0, ±6).

Since the foci are on the

*y*-axis, the major axis is along the*y*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly, 2

*b*= 16 ⇒*b*= 8 and*c*= 6.
It is known that.

Thus, the equation of the ellipse is.

#### Question 17:

Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0),

*a*= 4#### Answer:

Foci (±3, 0),

*a*= 4
Since the foci are on the

*x*-axis, the major axis is along the*x*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly,

*c*= 3 and*a*= 4.
It is known that.

Thus, the equation of the ellipse is.

#### Question 18:

Find the equation for the ellipse that satisfies the given conditions:

*b*= 3,*c*= 4, centre at the origin; foci on the*x*axis.#### Answer:

It is given that

*b*= 3,*c*= 4, centre at the origin; foci on the*x*axis.
Since the foci are on the

*x*-axis, the major axis is along the*x*-axis.
Therefore, the equation of the ellipse will be of the form, where

*a*is the semi-major axis.
Accordingly,

*b*= 3,*c*= 4.
It is known that.

Thus, the equation of the ellipse is.

#### Question 19:

Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the

*y*-axis and passes through the points (3, 2) and (1, 6).#### Answer:

Since the centre is at (0, 0) and the major axis is on the

*y*-axis, the equation of the ellipse will be of the form
The ellipse passes through points (3, 2) and (1, 6). Hence,

On solving equations (2) and (3), we obtain

*b*^{2}= 10 and*a*^{2}= 40.
Thus, the equation of the ellipse is.

#### Question 20:

Find the equation for the ellipse that satisfies the given conditions: Major axis on the

*x*-axis and passes through the points (4, 3) and (6, 2).#### Answer:

Since the major axis is on the

*x*-axis, the equation of the ellipse will be of the form
The ellipse passes through points (4, 3) and (6, 2). Hence,

On solving equations (2) and (3), we obtain

*a*^{2}= 52 and*b*^{2}= 13.
Thus, the equation of the ellipse is.

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