# NCERT solution class 11 chapter 11 Conic Sections exercise 11.3 mathematics

## EXERCISE 11.3

#### 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

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 = 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 = 2a = 12

Length of minor axis = 2b = 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

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 = 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 = 2a = 10

Length of minor axis = 2b = 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

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 = 4 and b = 3.

Therefore,

The coordinates of the foci are.

The coordinates of the vertices are.

Length of major axis = 2a = 8

Length of minor axis = 2b = 6

Length of latus rectum

#### Question 4:

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

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 = 5 and a = 10.

Therefore,

The coordinates of the foci are.

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

Length of major axis = 2a = 20

Length of minor axis = 2b = 10

Length of latus rectum

#### Question 5:

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

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 = 7 and b = 6.

Therefore,

The coordinates of the foci are.

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

Length of major axis = 2a = 14

Length of minor axis = 2b = 12

Length of latus rectum

#### Question 6:

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

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 = 10 and a = 20.

Therefore,

The coordinates of the foci are.

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

Length of major axis = 2a = 40

Length of minor axis = 2b = 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 36x2 + 4y2 = 144

The given equation is 36x2 + 4y2 = 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 = 2 and a = 6.

Therefore,

The coordinates of the foci are.

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

Length of major axis = 2= 12

Length of minor axis = 2b = 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 16x2 + y2 = 16

The given equation is 16x2 + y2 = 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 = 1 and a = 4.

Therefore,

The coordinates of the foci are.

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

Length of major axis = 2a = 8

Length of minor axis = 2b = 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 4x2 + 9y2 = 36

The given equation is 4x2 + 9y2 = 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 = 3 and b = 2.

Therefore,

The coordinates of the foci are.

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

Length of major axis = 2a = 6

Length of minor axis = 2b = 4

Length of latus rectum

#### Question 10:

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

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 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)

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 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)

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 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)

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 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)

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 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)

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 is the semi-major axis.

Accordingly, 2a = 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)

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 is the semi-major axis.

Accordingly, 2b = 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

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 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 axis.

It is given that b = 3, c = 4, centre at the origin; foci on the 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 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).

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 b2 = 10 and a2 = 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).