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NCERT solution class 11 chapter 11 Conic Sections exercise 11.3 mathematics

EXERCISE 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 = 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 

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 = 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 

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 = 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 

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 = 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 

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 = 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 

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 = 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

Answer:

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

Answer:

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

Answer:

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)

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 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 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 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 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 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 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)

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 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

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 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.

Answer:

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).

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 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).

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 a2 = 52 and b2 = 13.

Thus, the equation of the ellipse is.


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