The solutions for the questions present in the third exercise of Chapter 11, Class 11, maths are given here. These PDFs can be downloaded from the links given below. Exercise 11.3 of NCERT Solutions for Class 11 Maths Chapter 11- Conic Sections is based on the following topics:

- Ellipse
- Relationship between semi-major axis, semi-minor axis and the distance of the focus from the centre of the ellipse
- Special cases of an ellipse
- Eccentricity
- Standard equations of an ellipse
- Latus rectum

These solutions are prepared by subject matter experts at BYJUâ€™S, describing the complete method of solving problems. By understanding the concepts used in NCERT Solutions for Class 11 Maths, students will be able to clear all their doubts related to topics present in Class 11 maths.

### Download PDF of NCERT Solutions for Class 11 Maths Chapter 11- Conic Sections Exercise 11.3

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### Solutions for Class 11 Maths Chapter 11 â€“ Exercise 11.3

**In each of the Exercises 1 to 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.**

**1. x ^{2}/36 + y^{2}/16 = 1**

**Solution:**

Given:

The equation is x^{2}/36 + y^{2}/16 = 1

Here, the denominator of x^{2}/36Â is greater than the denominator of y^{2}/16.

So, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation withÂ x^{2}/a^{2} + y^{2}/b^{2} = 1, we get

a = 6 and b = 4.

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(36-16)

= âˆš20

= 2âˆš5

Then,

The coordinates of the foci are (2âˆš5, 0) and (-2âˆš5, 0).

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

Length of major axis = 2a = 2 (6) = 12

Length of minor axis = 2b = 2 (4) = 8

Eccentricity, eÂ = c/aÂ = 2âˆš5/6 = âˆš5/3

Length of latus rectum = 2b^{2}/a = (2Ã—16)/6 = 16/3

**2. x ^{2}/4 + y^{2}/25 = 1**

**Solution:**

Given:

The equation is x^{2}/4 + y^{2}/25 = 1

Here, the denominator of y^{2}/25Â is greater than the denominator of x^{2}/4.

So, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation with x^{2}/a^{2} + y^{2}/b^{2} = 1, we get

a = 5 and b = 2.

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(25-4)

= âˆš21

Then,

The coordinates of the foci are (0, âˆš21) and (0, -âˆš21).

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

Length of major axis = 2a = 2 (5) = 10

Length of minor axis = 2b = 2 (2) = 4

Eccentricity, e = c/aÂ = âˆš21/5

Length of latus rectum = 2b^{2}/a = (2Ã—2^{2})/5 = (2Ã—4)/5 = 8/5

**3. Â x ^{2}/16 + y^{2}/9 = 1**

**Solution:**

Given:

The equation is x^{2}/16 + y^{2}/9 = 1 or x^{2}/4^{2} + y^{2}/3^{2} = 1

Here, the denominator of x^{2}/16Â is greater than the denominator of y^{2}/9.

So, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation withÂ x^{2}/a^{2} + y^{2}/b^{2} = 1, we get

a = 4 and b = 3.

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(16-9)

= âˆš7

Then,

The coordinates of the foci are (âˆš7, 0) and (-âˆš7, 0).

The coordinates of the vertices are (4, 0) and (-4, 0)

Length of major axis = 2a = 2 (4) = 8

Length of minor axis = 2b = 2 (3) = 6

Eccentricity, e = c/aÂ = âˆš7/4

Length of latus rectum = 2b^{2}/a = (2Ã—3^{2})/4 = (2Ã—9)/4 = 18/4 = 9/2

**4. Â x ^{2}/25 + y^{2}/100 = 1**

**Solution:**

Given:

The equation is x^{2}/25 + y^{2}/100 = 1

Here, the denominator of y^{2}/100Â is greater than the denominator of x^{2}/25.

So, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing the given equation withÂ x^{2}/b^{2} + y^{2}/a^{2} = 1, we get

b = 5 and a =10.

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(100-25)

= âˆš75

= 5âˆš3

Then,

The coordinates of the foci are (0, 5âˆš3) and (0, -5âˆš3).

The coordinates of the vertices are (0, âˆš10) and (0, -âˆš10)

Length of major axis = 2a = 2 (10) = 20

Length of minor axis = 2b = 2 (5) = 10

Eccentricity, e = c/aÂ = 5âˆš3/10 = âˆš3/2

Length of latus rectum = 2b^{2}/a = (2Ã—5^{2})/10 = (2Ã—25)/10 = 5

**5. x ^{2}/49 + y^{2}/36 = 1**

**Solution:**

Given:

The equation is x^{2}/49 + y^{2}/36 = 1

Here, the denominator of x^{2}/49Â is greater than the denominator of y^{2}/36.

So, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation withÂ x^{2}/a^{2} + y^{2}/b^{2} = 1, we get

b = 6 and a =7

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(49-36)

= âˆš13

Then,

The coordinates of the foci are (âˆš13, 0) and (-âˆš3, 0).

The coordinates of the vertices are (7, 0) and (-7, 0)

Length of major axis = 2a = 2 (7) = 14

Length of minor axis = 2b = 2 (6) = 12

Eccentricity, e = c/aÂ = âˆš13/7

Length of latus rectum = 2b^{2}/a = (2Ã—6^{2})/7 = (2Ã—36)/7 = 72/7

**6. x ^{2}/100 + y^{2}/400 = 1**

**Solution:**

Given:

The equation is x^{2}/100 + y^{2}/400 = 1

Here, the denominator of y^{2}/400Â is greater than the denominator of x^{2}/100.

So, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing the given equation withÂ x^{2}/b^{2} + y^{2}/a^{2} = 1, we get

b = 10 and a =20.

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(400-100)

= âˆš300

= 10âˆš3

Then,

The coordinates of the foci are (0, 10âˆš3) and (0, -10âˆš3).

The coordinates of the vertices are (0, 20) and (0, -20)

Length of major axis = 2a = 2 (20) = 40

Length of minor axis = 2b = 2 (10) = 20

Eccentricity, e = c/aÂ = 10âˆš3/20 = âˆš3/2

Length of latus rectum = 2b^{2}/a = (2Ã—10^{2})/20 = (2Ã—100)/20 = 10

**7. 36x ^{2}Â + 4y^{2}Â = 144**

**Solution:**

Given:

The equation is 36x^{2}Â + 4y^{2}Â = 144 or x^{2}/4 + y^{2}/36 = 1 or x^{2}/2^{2} + y^{2}/6^{2} = 1

Here, the denominator of y^{2}/6^{2}Â is greater than the denominator of x^{2}/2^{2}.

So, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing the given equation withÂ x^{2}/b^{2} + y^{2}/a^{2} = 1, we get

b = 2 and a = 6.

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(36-4)

= âˆš32

= 4âˆš2

Then,

The coordinates of the foci are (0, 4âˆš2) and (0, -4âˆš2).

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

Length of major axis = 2a = 2 (6) = 12

Length of minor axis = 2b = 2 (2) = 4

Eccentricity, e = c/aÂ = 4âˆš2/6 = 2âˆš2/3

Length of latus rectum = 2b^{2}/a = (2Ã—2^{2})/6 = (2Ã—4)/6 = 4/3

**8. 16x ^{2}Â + y^{2}Â = 16**

**Solution:**

Given:

The equation is 16x^{2}Â + y^{2}Â = 16 or x^{2}/1 + y^{2}/16 = 1 or x^{2}/1^{2} + y^{2}/4^{2} = 1

Here, the denominator of y^{2}/4^{2}Â is greater than the denominator of x^{2}/1^{2}.

So, the major axis is along the y-axis, while the minor axis is along the x-axis.

On comparing the given equation withÂ x^{2}/b^{2} + y^{2}/a^{2} = 1, we get

b =1 and a =4.

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(16-1)

= âˆš15

Then,

The coordinates of the foci are (0, âˆš15) and (0, -âˆš15).

The coordinates of the vertices are (0, 4) and (0, -4)

Length of major axis = 2a = 2 (4) = 8

Length of minor axis = 2b = 2 (1) = 2

Eccentricity, e = c/aÂ = âˆš15/4

Length of latus rectum = 2b^{2}/a = (2Ã—1^{2})/4 = 2/4 = Â½

**9. 4x ^{2}Â + 9y^{2}Â = 36**

**Solution:**

Given:

The equation is 4x^{2}Â + 9y^{2}Â = 36 or x^{2}/9 + y^{2}/4 = 1 or x^{2}/3^{2} + y^{2}/2^{2} = 1

Here, the denominator of x^{2}/3^{2}Â is greater than the denominator of y^{2}/2^{2}.

So, the major axis is along the x-axis, while the minor axis is along the y-axis.

On comparing the given equation withÂ x^{2}/a^{2} + y^{2}/b^{2} = 1, we get

a =3 and b =2.

c = âˆš(a^{2}Â â€“ b^{2})

= âˆš(9-4)

= âˆš5

Then,

The coordinates of the foci are (âˆš5, 0) and (-âˆš5, 0).

The coordinates of the vertices are (3, 0) and (-3, 0)

Length of major axis = 2a = 2 (3) = 6

Length of minor axis = 2b = 2 (2) = 4

Eccentricity, e = c/aÂ = âˆš5/3

Length of latus rectum = 2b^{2}/a = (2Ã—2^{2})/3 = (2Ã—4)/3 = 8/3

**In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions:**

**10. Vertices (Â±Â 5, 0), foci (Â±Â 4, 0)**

**Solution:**

Given:

Vertices (Â±Â 5, 0) and foci (Â±Â 4, 0)

Here, the vertices are on the x-axis.

So, the equation of the ellipse will be of the formÂ x^{2}/a^{2} + y^{2}/b^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, a = 5 and c = 4.

It is known that a^{2}Â = b^{2 }+ c^{2}.

So, 5^{2}Â = b^{2 }+ 4^{2}

25 = b^{2}Â + 16

b^{2}Â = 25 â€“ 16

b =Â âˆš9

= 3

âˆ´ The equation of the ellipse is x^{2}/5^{2} + y^{2}/3^{2} = 1 or x^{2}/25 + y^{2}/9Â = 1

**11. Vertices (0,Â Â±Â 13), foci (0,Â Â± 5)**

**Solution:**

Given:

Vertices (0,Â Â±Â 13) and foci (0,Â Â± 5)

Here, the vertices are on the y-axis.

So, the equation of the ellipse will be of the formÂ x^{2}/b^{2} + y^{2}/a^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, a =13 and c = 5.

It is known that a^{2}Â = b^{2 }+ c^{2}.

13^{2}Â = b^{2}+5^{2}

169 = b^{2}Â + 15

b^{2}Â = 169 â€“ 125

b =Â âˆš144

= 12

âˆ´ The equation of the ellipse is x^{2}/12^{2} + y^{2}/13^{2}Â = 1 or x^{2}/144 + y^{2}/169Â = 1

**12. Vertices (Â±Â 6, 0), foci (Â± 4, 0)**

**Solution:**

Given:

Vertices (Â±Â 6, 0) and foci (Â± 4, 0)

Here, the vertices are on the x-axis.

So, the equation of the ellipse will be of the formÂ x^{2}/a^{2} + y^{2}/b^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, a = 6 and c = 4.

It is known that a^{2}Â = b^{2 }+ c^{2}.

6^{2}Â = b^{2}+4^{2}

36 = b^{2}Â + 16

b^{2}Â = 36 â€“ 16

b =Â âˆš20

âˆ´ The equation of the ellipse is x^{2}/6^{2} + y^{2}/(âˆš20)^{2}Â = 1 or x^{2}/36 + y^{2}/20Â = 1

**13. Ends of major axis (Â± 3, 0), ends of minor axis (0,Â Â±2)**

**Solution:**

Given:

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

Here, the major axis is along the x-axis.

So, the equation of the ellipse will be of the formÂ x^{2}/a^{2} + y^{2}/b^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, a = 3 and b = 2.

âˆ´ The equation for the ellipse x^{2}/3^{2} + y^{2}/2^{2}Â = 1 or x^{2}/9 + y^{2}/4Â = 1

**14. Ends of major axis (0,Â Â±âˆš5), ends of minor axis (Â±1, 0)**

**Solution:**

Given:

Ends of major axis (0,Â Â±âˆš5) and ends of minor axis (Â±1, 0)

Here, the major axis is along the y-axis.

So, the equation of the ellipse will be of the formÂ x^{2}/b^{2} + y^{2}/a^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, a =Â âˆš5 and b = 1.

âˆ´ The equation for the ellipse x^{2}/1^{2} + y^{2}/(âˆš5)^{2}Â = 1 or x^{2}/1 + y^{2}/5Â = 1

**15. Length of major axis 26, foci (Â±5, 0)**

**Solution:**

Given:

Length of major axis is 26 and foci (Â±5, 0)

Since the foci are on the x-axis, the major axis is along the x-axis.

^{2}/a^{2} + y^{2}/b^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, 2a = 26

a = 13 and c = 5.

It is known that a^{2}Â = b^{2 }+ c^{2}.

13^{2}Â = b^{2}+5^{2}

169 = b^{2}Â + 25

b^{2}Â = 169 â€“ 25

b =Â âˆš144

= 12

âˆ´ The equation of the ellipse is x^{2}/13^{2} + y^{2}/12^{2} = 1Â or x^{2}/169 + y^{2}/144Â = 1

**16. Length of minor axis 16, foci (0,Â Â±6).**

**Solution:**

Given:

Length of minor axis is 16 and foci (0,Â Â±6).

Since the foci are on the y-axis, the major axis is along the y-axis.

So, the equation of the ellipse will be of the formÂ x^{2}/b^{2} + y^{2}/a^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, 2b =16

b = 8 and c = 6.

It is known that a^{2}Â = b^{2 }+ c^{2}.

a^{2}Â = 8^{2 }+ 6^{2}

=Â 64 + 36

=100

a =Â âˆš100

= 10

âˆ´ The equation of the ellipse is x^{2}/8^{2} + y^{2}/10^{2}Â =1 or x^{2}/64 + y^{2}/100Â = 1

**17. Foci (Â±3, 0), a = 4**

**Solution:**

Given:

Foci (Â±3, 0) and a = 4

Since the foci are on the x-axis, the major axis is along the x-axis.

So, the equation of the ellipse will be of the form x^{2}/a^{2} + y^{2}/b^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, c = 3 and a = 4.

It is known that a^{2}Â = b^{2 }+ c^{2}.

a^{2}Â = 8^{2 }+ 6^{2}

= 64 + 36

= 100

16 = b^{2}Â + 9

b^{2}Â = 16 â€“ 9

= 7

âˆ´ The equation of the ellipse is x^{2}/16 + y^{2}/7 = 1

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

**Solution:**

Given:

b = 3, c = 4, centre at the origin and foci on the x axis.

Since the foci are on the x-axis, the major axis is along the x-axis.

^{2}/a^{2} + y^{2}/b^{2} = 1, where â€˜aâ€™ is the semi-major axis.

Then, b = 3 and c = 4.

It is known that a^{2}Â = b^{2 }+ c^{2}.

a^{2}Â = 3^{2 }+ 4^{2}

= 9 + 16

=25

a = âˆš25

= 5

âˆ´ The equation of the ellipse is x^{2}/5^{2} + y^{2}/3^{2} or x^{2}/25 + y^{2}/9 = 1

**19. Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).**

**Solution:**

Given:

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 x^{2}/b^{2} + y^{2}/a^{2} = 1, where â€˜aâ€™ is the semi-major axis.

The ellipse passes through points (3, 2) and (1, 6).

So, by putting the values x = 3 and y = 2, we get,

3^{2}/b^{2} + 2^{2}/a^{2} = 1

9/b^{2} + 4/a^{2}â€¦. (1)

And by putting the values x = 1 and y = 6, we get,

1^{1}/b^{2} + 6^{2}/a^{2} = 1

1/b^{2} + 36/a^{2} = 1 â€¦. (2)

On solving equation (1) and (2), we get

b^{2}Â = 10 and a^{2}Â = 40.

âˆ´ The equation of the ellipse is x^{2}/10 + y^{2}/40Â = 1 or 4x^{2}Â + yÂ ^{2}Â = 40

**20. Major axis on the x-axis and passes through the points (4,3) and (6,2).**

**Solution:**

Given:

Major axis on the x-axis and passes through the points (4, 3) and (6, 2).

Since the major axis is on the x-axis, the equation of the ellipse will be the form

x^{2}/a^{2} + y^{2}/b^{2} = 1â€¦. (1) [Where â€˜aâ€™ is the semi-major axis.]

The ellipse passes through points (4, 3) and (6, 2).

So by putting the values x = 4 and y = 3 in equation (1), we get,

16/a^{2} + 9/b^{2} = 1 â€¦. (2)

Putting, x = 6 and y = 2 in equation (1), we get,

36/a^{2} + 4/b^{2} = 1 â€¦. (3)

From equation (2)

16/a^{2} = 1 â€“ 9/b^{2}

1/a^{2} = (1/16 (1 â€“ 9/b^{2})) â€¦. (4)

Substituting the value of 1/a^{2} in equation (3) we get,

36/a^{2} + 4/b^{2} = 1

36(1/a^{2}) + 4/b^{2} = 1

36[1/16 (1 â€“ 9/b^{2})] + 4/b^{2} = 1

36/16 (1 â€“ 9/b^{2}) + 4/b^{2} = 1

9/4 (1 â€“ 9/b^{2}) + 4/b^{2} = 1

9/4 â€“ 81/4b^{2} + 4/b^{2} = 1

-81/4b^{2} + 4/b^{2} = 1 â€“ 9/4

(-81+16)/4b^{2} = (4-9)/4

-65/4b^{2} = -5/4

-5/4(13/b^{2}) = -5/4

13/b^{2} = 1

1/b^{2} = 1/13

b^{2} = 13

Now substitute the value of b^{2} in equation (4) we get,

1/a^{2} = 1/16(1 â€“ 9/b^{2})

= 1/16(1 â€“ 9/13)

= 1/16((13-9)/13)

= 1/16(4/13)

= 1/52

a^{2} = 52

Equation of ellipse is x^{2}/a^{2} + y^{2}/b^{2} = 1

By substituting the values of a^{2} and b^{2} in above equation we get,

x^{2}/52 + y^{2}/13 = 1

### Access Other Exercise Solutions of Class 11 Maths Chapter 11- Conic Sections

Exercise 11.1 Solutions 15 Questions

Exercise 11.2 Solutions 12 Questions

Exercise 11.4 Solutions 15 Questions

Miscellaneous Exercise On Chapter 11 Solutions 8 Questions