**NCERT solutions of class 11 maths exercise 11.3 chapter 11-Conic Section**

Here all NCERT Solutions of Class 11 Maths Exercise 11.3 Chapter 11 Are Solved For Helping The Students of Class 11 In Clearing The Concept Of Chapter 11 Conic Section Completely, So That They Could Attemp All The Questions Of The Chapter 11 Conic Section Mentioned In The Maths Question Paper Of CBSE Board Class 11.

**Download pdf of NCERT solutions class 11 chapter 11-Conic Section**

**pdf-NCERT solutions class 11 chapter 11-Conic Section**

**NCERT solutions of class 11 maths**

Chapter 1-Sets | Chapter 9-Sequences and Series |

Chapter 2- Relations and functions | Chapter 10- Straight Lines |

Chapter 3- Trigonometry | Chapter 11-Conic Sections |

Chapter 4-Principle of mathematical induction | Chapter 12-Introduction to three Dimensional Geometry |

Chapter 5-Complex numbers | Chapter 13- Limits and Derivatives |

Chapter 6- Linear Inequalities | Chapter 14-Mathematical Reasoning |

Chapter 7- Permutations and Combinations | Chapter 15- Statistics |

Chapter 8- Binomial Theorem | Chapter 16- Probability |

**CBSE Class 11-Question paper of maths 2015**

**CBSE Class 11 – Second unit test of maths 2021 with solutions**

**Study notes of Maths and Science NCERT and CBSE from class 9 to 12**

**NCERT solutions of class 11 maths exercise 11.3 chapter 11-Conic Section**

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 length of the latus rectum of the ellipse.

Ans. The given equation of the ellipse is

The equation of ellipse shows that major axis of the ellipse is along x-axis

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

a² = 36 ⇒ a = ±6 and b² = 16⇒b =±4

The coordinates of the foci are (±c,0)

c = √(a²-b²) = √(6² -4²) =√(36 -16) =√20

Therefore the coordinates of the foci are (±√20,0)

The vertices of the ellipse are (±a,0) = (±6,0)

The length of major axis is = 2a = 2× 6 = 12

The length of minor axis = 2b = 2×4 = 8

The eccentricity of ellipse is = e =c/a = √20/6

Length of the latus rectum of the ellipse = 2b²/a =2×4²/6=16/3

Ans. The given equation of the ellipse is

The equation of ellipse shows that major axis of the ellipse is along y-axis(25>4)

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

b² = 4 ⇒ b = ±2 and a² = 25⇒a =±5

The coordinates of the foci are (±c,0)

c = √(a²-b²) = √(5² -2²) =√(25 -4) =√21

Therefore the coordinates of the foci are (0,±√21)

The vertices of the ellipse are (0,±a) = (0,±5)

The length of major axis is = 2a = 2× 5 = 10

The length of minor axis = 2b = 2×2 = 4

The eccentricity of ellipse is = e =c/a = √21/5

Length of the latus rectum of the ellipse = 2b²/a =2×2²/5=8/5

Ans. The given equation of the ellipse is

The equation of ellipse shows that major axis of the ellipse is along x-axis(16>9)

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

a² = 16 ⇒ a = ±4 and b² = 9⇒b=±3

The coordinates of the foci are (±c,0)

c = √(a²-b²) = √(4² -3²) =√(16 -9) =√7

Therefore the coordinates of the foci are (±√7,0)

The vertices of the ellipse are (±a,0) = (±4,0)

The length of major axis is = 2a = 2× 4 = 8

The length of minor axis = 2b = 2×3 = 6

The eccentricity of ellipse is = e =c/a = √7/4

Length of the latus rectum of the ellipse = 2b²/a =2×3²/4=9/2

Ans. The given equation of the ellipse is

The equation of ellipse shows that major axis of the ellipse is along y-axis(100>25)

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

a² = 100⇒ a = ±10 and b² = 25⇒b=±5

The coordinates of the foci are (0,±c)

c = √(a²-b²) = √(10² -5²) =√(100 -25) =√75= 5√3

Therefore the coordinates of the foci are (0,±5√3)

The vertices of the ellipse are (0,±a) = (0,±10)

The length of major axis is = 2a = 2× 10= 20

The length of minor axis = 2b = 2×5 = 10

The eccentricity of ellipse is = e =c/a = 5√3/10 =√3/2

Length of the latus rectum of the ellipse = 2b²/a =2×(5)²/10=5

**NCERT solutions of class 11 maths exercise 11.3 chapter 11-Conic Section**

Ans. The given equation of the ellipse is

The equation of ellipse shows that major axis of the ellipse is along y-axis(49>36)

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

a² = 49⇒ a = ±7 and b² = 36⇒b=±6

The coordinates of the foci are (±c,0)

c = √(a²-b²) = √(7² -6²) =√(49-36) =√13

Therefore the coordinates of the foci are (±√13,0)

The vertices of the ellipse are (±a,0) = (±7,0)

The length of major axis is = 2a = 2× 7= 14

The length of minor axis = 2b = 2×6= 12

The eccentricity of ellipse is = e =c/a = √13/7 =√13/7

Length of the latus rectum of the ellipse = 2b²/a =2×(6)²/7=72/7

Ans. The given equation of the ellipse is

The equation of ellipse shows that major axis of the ellipse is along y-axis (400>100)

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

a² = 400⇒ a = ±20 and b² = 100⇒b=±10

The coordinates of the foci are (0,±c)

c = √(a²-b²) = √(400 -100) =√300 = 10√3

Therefore the coordinates of the foci are (0,±10√3)

The vertices of the ellipse are (0,±a) = (0,±20)

The length of major axis is = 2a = 2×20 = 40

The length of minor axis = 2b = 2×10= 20

The eccentricity of ellipse is = e =c/a = 10√3/20 =√3/2

Length of the latus rectum of the ellipse = 2b²/a =2×(10)²/20=10

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Ans. The given equation of the ellipse is

Rearranging the equation as follows

The equation of ellipse shows that major axis of the ellipse is along y-axis (36>4)

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

a² = 36⇒ a = ±6 and b² = 4⇒b=±2

The coordinates of the foci are (0,±c)

c = √(a²-b²) = √(6² -2²) =√(36-4) =√32= 4√2

Therefore the coordinates of the foci are (0,±4√2)

The vertices of the ellipse are (0,±a) = (0,±6)

The length of major axis is = 2a = 2× 6= 12

The length of minor axis = 2b = 2×2= 4

The eccentricity of ellipse is = e =c/a = 4√2/6 =2√2/3

Length of the latus rectum of the ellipse = 2b²/a =2×(2)²/6=4/3

Ans. The given equation of the ellipse is

Rearranging the equation as follows

The equation of ellipse shows that major axis of the ellipse is along y-axis (16>1)

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

a² = 16⇒ a = ±4 and b² = 1⇒b=±1

The coordinates of the foci are (0,±c)

c = √(a²-b²) = √(4² -1²) =√(16-1) =√15

Therefore the coordinates of the foci are (0,±√15)

The vertices of the ellipse are (0,±a) = (0,±4)

The length of major axis is = 2a = 2× 4= 8

The length of minor axis = 2b = 2×1= 2

The eccentricity of ellipse is = e =c/a = √15/4

Length of the latus rectum of the ellipse = 2b²/a =2×(1)²/4=1/2

Ans. The given equation of the ellipse is

Rearranging the equation as follows

The equation of ellipse shows that major axis of the ellipse is along y-axis (9>4)

The standard equation of ellipse is

On Comparing the given equation with the standard equation,we get

a² = 9⇒ a = ±3 and b² = 4⇒b=±2

The coordinates of the foci are (±c,0)

c = √(a²-b²) = √(3² -2²) =√(9-4) =√5

Therefore the coordinates of the foci are (±√5,0)

The vertices of the ellipse are (±a,0) = (±3,0)

The length of major axis is = 2a = 2× 3= 6

The length of minor axis = 2b = 2×2= 4

The eccentricity of ellipse is = e =c/a = √5/3

Length of the latus rectum of the ellipse = 2b²/a =2×(2)²/3=8/3

**In each of the exercises 10 to 20,find the equations for the ellipse that satisfies the given conditions.**

**NCERT solutions of class 11 maths exercise 11.3 chapter 11-Conic Section**

Ans. We are given the vertices (±a,0) = (±5,0), foci (±c,0) = (±4,0)

So,a = 5, c = 4

Since c² = a² – b² ⇒b =±√(a² -c²) =±√(5² – 4²) = ±√(25 -16) =±√9=±3

The equation of equation of ellipse is

Putting the value of a = 5 and b =3 in the equation

**Q11. Vertices (0, ±13), foci (0, ± 5)**

Ans. We are given the vertices (0,±a) = (0,±13), foci (0, ±c) = (0,±5)

So,a = 13, c = 5

Since c² = a² – b² ⇒b =±√(a² -c²) =±√(13² – 5²) = ±√(169 -25) =±√144=±12

The equation of equation of ellipse is

Putting the value of a = 13 and b =12 in the equation

**Q12. Vertices (±6,0), foci ( ± 4,0)**

Ans. We are given the vertices (±a,0) = (±6,0), foci ( ±c,0) = (±4,0)

So,a = 6, c = 4

Since c² = a² – b² ⇒b =±√(a² -c²) =±√(6² – 4²) = ±√(36 -16) =±√20=±2√5

The equation of ellipse is

Putting the value of a = 6 and b =2√5 in the equation

**Q13. Ends of the major axis (±3,0), ends of the minor axis (0, ± 2)**

Ans. We are given the vertices i.e ends of the major axis (±a,0) = (±3,0), ends of the minor axis i.e (0,b) = (0, ± 2)

So,a = 3, b= 2

The equation of equation of ellipse is

**Q14. Ends of the major axis (0,±√5), ends of the minor axis (± 1,0)**

Ans. We are given the vertices i.e ends of the major axis (0, ±a) = (0,±√5), ends of the minor axis i.e (±b, 0) = (± 1,0)

So,a = √5, b= 1

The equation of the ellipse is

**Q15. Length of major axis 26, foci (±5, 0)**

**Ans.**We are given length of major axis 26, foci (±5, 0)

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

Length of major axis, 2a = 26 ⇒ a = 13

Foci, (±c,0) = (±5, 0)⇒ c = 5

It is known that a^{2} = b^{2 }+ c^{2}.⇒b =±√(a²-c²) = ±√(13² – 5²)= ±√(169 – 25)=±√144=±12⇒b =12

The equation of the ellipse is

Putting the value of a =13, b = 12

**NCERT solutions of class 11 maths exercise 11.3 chapter 11-Conic Section**

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

**Ans.**We are given length of minor axis 16, foci (0,±6)

Length of minor axis, 2b = 16 ⇒ b = 8

Foci, (0, ±c) = (0, ±6)⇒ c = 6

It is known that a^{2} = b^{2 }+ c^{2}.⇒a =±√(b²+c²) = ±√(8² + 6²)= ±√(64 + 36)=±√100=±10⇒a =10

The equation of the ellipse is

Putting the value a =10,b =8

∴ The equation of the ellipse

**Q17.Foci (±3, 0), a = 4**

Ans. We are given that

Foci (±3, 0) and a = 4

Foci, ( ±c,0) = ( ±3,0)⇒ c = 3

It is known that a^{2} = b^{2 }+ c^{2}.⇒b =±√(a²-c²) = ±√(4² – 3²)= ±√(16- 9)=±√7=±√7⇒a =√7

The equation of ellipse is

Putting b = √7 and a = 4

**Q18. b = 3, c = 4, centre at the origin; foci on the x axis.**

**Ans. **We are given that

b = 3, c = 4, centre is(0,0) and foci on the x axis =(±c,0) = (±3,0)

It is known that a² = b² + c² ⇒ a =±√(b² +c²) = ±√(3² +4²) =±√25=±5

The equation of the ellipse is

Putting the value of a =5 and b =3

**Q19. Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).**

Ans. We are given centre of ellipse at (0,0) and it is passing through (3, 2) and (1, 6)

Major axis is on y-axis,so the eqation is

Putting the value of (x,y)= (3, 2) and (x,y) =(1, 6) in the equation,we get the equation (i) and equation (ii)

Solving (i) and (ii) equations

b² = 10 and a²= 40.

Substituting the value of b² = 10 and a²= 40 in the equation,we get the required equation

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

Ans. We are given that the ellipse is passing through (4, 3) and ( 6,2)

Major axis is on x-axis,so the eqation is

Putting the value of (x,y)= (4, 3) and (x,y) =(6, 2) in the equation,we get the equation (i) and equation (ii)

Solving (i) and (ii) equations

b² = 13 and a²= 52.

Substituting the value of b² = 13 and a²= 52 in the equation,we get the required equation

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Chapter 1- Number System | Chapter 9-Areas of parallelogram and triangles |

Chapter 2-Polynomial | Chapter 10-Circles |

Chapter 3- Coordinate Geometry | Chapter 11-Construction |

Chapter 4- Linear equations in two variables | Chapter 12-Heron’s Formula |

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**NCERT Solutions of class 9 science **

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**NCERT solutions of class 11 maths**

Chapter 1-Sets | Chapter 9-Sequences and Series |

Chapter 2- Relations and functions | Chapter 10- Straight Lines |

Chapter 3- Trigonometry | Chapter 11-Conic Sections |

Chapter 4-Principle of mathematical induction | Chapter 12-Introduction to three Dimensional Geometry |

Chapter 5-Complex numbers | Chapter 13- Limits and Derivatives |

Chapter 6- Linear Inequalities | Chapter 14-Mathematical Reasoning |

Chapter 7- Permutations and Combinations | Chapter 15- Statistics |

Chapter 8- Binomial Theorem | Chapter 16- Probability |

**CBSE Class 11-Question paper of maths 2015**

**CBSE Class 11 – Second unit test of maths 2021 with solutions**

**NCERT solutions of class 12 maths**

Chapter 1-Relations and Functions | Chapter 9-Differential Equations |

Chapter 2-Inverse Trigonometric Functions | Chapter 10-Vector Algebra |

Chapter 3-Matrices | Chapter 11 – Three Dimensional Geometry |

Chapter 4-Determinants | Chapter 12-Linear Programming |

Chapter 5- Continuity and Differentiability | Chapter 13-Probability |

Chapter 6- Application of Derivation | CBSE Class 12- Question paper of maths 2021 with solutions |

Chapter 7- Integrals | |

Chapter 8-Application of Integrals |

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