**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 Attempt All The Questions Of The Chapter 11 Conic Section Mentioned In The Maths Question Paper Of CBSE Board Class 11.

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

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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 standardย equation,we get

aยฒ = 36 โ a = ยฑ6 and bยฒ = 16โb =ยฑ4

The coordinates of the foci are (ยฑc,0)

c = โ(aยฒ-bยฒ) = โ(6ยฒ -4ยฒ) =โ(36 -16) =โ20

Therefore the coordinates of the foci are (ยฑโ20,0)

The vertices of the ellipse are (ยฑa,0) = (ยฑ6,0)

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

Theย length of minor axis = 2b = 2ร4 = 8

The eccentricity of ellipse is = e =c/a = โ20/6

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร4ยฒ/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 standardย equation,we get

bยฒ = 4 โ b = ยฑ2 and aยฒ = 25โa =ยฑ5

The coordinates of the foci are (ยฑc,0)

c = โ(aยฒ-bยฒ) = โ(5ยฒ -2ยฒ) =โ(25 -4) =โ21

Therefore the coordinates of the foci are (0,ยฑโ21)

The vertices of the ellipse are (0,ยฑa) = (0,ยฑ5)

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

Theย length of minor axis = 2b = 2ร2 = 4

The eccentricity of ellipse is = e =c/a = โ21/5

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร2ยฒ/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 standardย equation,we get

aยฒ = 16 โ a = ยฑ4ย and bยฒ = 9โb=ยฑ3

The coordinates of the foci are (ยฑc,0)

c = โ(aยฒ-bยฒ) = โ(4ยฒ -3ยฒ) =โ(16 -9) =โ7

Therefore the coordinates of the foci are (ยฑโ7,0)

The vertices of the ellipse are (ยฑa,0) = (ยฑ4,0)

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

Theย length of minor axis = 2b = 2ร3 = 6

The eccentricity of ellipse is = e =c/a = โ7/4

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร3ยฒ/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 standardย equation,we get

aยฒ = 100โ a = ยฑ10 and bยฒ = 25โb=ยฑ5

The coordinates of the foci are (0,ยฑc)

c = โ(aยฒ-bยฒ) = โ(10ยฒ -5ยฒ) =โ(100 -25) =โ75= 5โ3

Therefore the coordinates of the foci are (0,ยฑ5โ3)

The vertices of the ellipse are (0,ยฑa) = (0,ยฑ10)

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

Theย length of minor axis = 2b = 2ร5 = 10

The eccentricity of ellipse is = e =c/a = 5โ3/10 =โ3/2

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร(5)ยฒ/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 standardย equation,we get

aยฒ = 49โ a = ยฑ7 and bยฒ = 36โb=ยฑ6

The coordinates of the foci are (ยฑc,0)

c = โ(aยฒ-bยฒ) = โ(7ยฒ -6ยฒ) =โ(49-36) =โ13

Therefore the coordinates of the foci are (ยฑโ13,0)

The vertices of the ellipse are (ยฑa,0) = (ยฑ7,0)

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

Theย length of minor axis = 2b = 2ร6= 12

The eccentricity of ellipse is = e =c/a = โ13/7 =โ13/7

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร(6)ยฒ/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 standardย equation,we get

aยฒ = 400โ a = ยฑ20 and bยฒ = 100โb=ยฑ10

The coordinates of the foci are (0,ยฑc)

c = โ(aยฒ-bยฒ) = โ(400 -100) =โ300 = 10โ3

Therefore the coordinates of the foci are (0,ยฑ10โ3)

The vertices of the ellipse are (0,ยฑa) = (0,ยฑ20)

The length of major axis is = 2a = 2ร20 = 40

Theย length of minor axis = 2b = 2ร10= 20

The eccentricity of ellipse is = e =c/a = 10โ3/20ย =โ3/2

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร(10)ยฒ/20=10

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 standardย equation,we get

aยฒ = 36โ a = ยฑ6 and bยฒ = 4โb=ยฑ2

The coordinates of the foci are (0,ยฑc)

c = โ(aยฒ-bยฒ) = โ(6ยฒ -2ยฒ) =โ(36-4) =โ32= 4โ2

Therefore the coordinates of the foci are (0,ยฑ4โ2)

The vertices of the ellipse are (0,ยฑa) = (0,ยฑ6)

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

Theย length of minor axis = 2b = 2ร2= 4

The eccentricity of ellipse is = e =c/a = 4โ2/6 =2โ2/3

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร(2)ยฒ/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 standardย equation,we get

aยฒ = 16โ a = ยฑ4 and bยฒ = 1โb=ยฑ1

The coordinates of the foci are (0,ยฑc)

c = โ(aยฒ-bยฒ) = โ(4ยฒ -1ยฒ) =โ(16-1) =โ15

Therefore the coordinates of the foci are (0,ยฑโ15)

The vertices of the ellipse are (0,ยฑa) = (0,ยฑ4)

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

Theย length of minor axis = 2b = 2ร1= 2

The eccentricity of ellipse is = e =c/a = โ15/4

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร(1)ยฒ/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 standardย equation,we get

aยฒ = 9โ a = ยฑ3 and bยฒ = 4โb=ยฑ2

The coordinates of the foci are (ยฑc,0)

c = โ(aยฒ-bยฒ) = โ(3ยฒ -2ยฒ) =โ(9-4) =โ5

Therefore the coordinates of the foci are (ยฑโ5,0)

The vertices of the ellipse are (ยฑa,0) = (ยฑ3,0)

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

Theย length of minor axis = 2b = 2ร2= 4

The eccentricity of ellipse is = e =c/a = โ5/3

Length of the latus rectum of the ellipse = 2bยฒ/a =2ร(2)ยฒ/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 (ยฑa,0) = (ยฑ5,0), foci (ยฑc,0) = (ยฑ4,0)

So,a = 5, c = 4

Since cยฒ = aยฒ – bยฒ โb =ยฑโ(aยฒ -cยฒ) =ยฑโ(5ยฒ – 4ยฒ) = ยฑโ(25 -16) =ยฑโ9=ยฑ3

The equation of equation of ellipse is

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

**Q11. Vertices (0, ยฑ13), foci (0, ยฑ 5)**

Ans. We are given the vertices (0,ยฑa) = (0,ยฑ13), foci (0, ยฑc) = (0,ยฑ5)

So,a = 13, c = 5

Since cยฒ = aยฒ – bยฒ โb =ยฑโ(aยฒ -cยฒ) =ยฑโ(13ยฒ – 5ยฒ) = ยฑโ(169 -25) =ยฑโ144=ยฑ12

The equation of equation of ellipse is

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

**Q12. Vertices (ยฑ6,0), foci ( ยฑ 4,0)**

Ans. We are given the vertices (ยฑa,0) = (ยฑ6,0), foci ( ยฑc,0) = (ยฑ4,0)

So,a = 6, c = 4

Since cยฒ = aยฒ – bยฒ โb =ยฑโ(aยฒ -cยฒ) =ยฑโ(6ยฒ – 4ยฒ) = ยฑโ(36 -16) =ยฑโ20=ยฑ2โ5

The equationย of ellipse is

Putting the value of a = 6 and b =2โ5ย in the equation

**Q13. Ends of the major axis (ยฑ3,0), ends of the minor axis (0, ยฑ 2)**

Ans. We are given the vertices i.e ends of the major axis (ยฑa,0) = (ยฑ3,0), ends of the minor axis i.e (0,b) = (0, ยฑ 2)

So,a = 3, b= 2

The equation of equation of ellipse is

**Q14. Ends of the major axis (0,ยฑโ5), ends of the minor axis (ยฑ 1,0)**

Ans. We are given the vertices i.e ends of the major axis (0, ยฑa) = (0,ยฑโ5), ends of the minor axis i.e (ยฑb, 0) = (ยฑ 1,0)

So,a = โ5, b= 1

The equationย ofย the ellipse is

**Q15. Length of major axis 26, foci (ยฑ5, 0)**

**Ans.**We are given length of major axis 26, foci (ยฑ5, 0)

Length of major axis is 26 and foci (ยฑ5, 0)

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

Foci, (ยฑc,0) = (ยฑ5, 0)โ c = 5

It is known that a^{2}ย = b^{2ย }+ c^{2}.โb =ยฑโ(aยฒ-cยฒ) = ยฑโ(13ยฒ – 5ยฒ)= ยฑโ(169 – 25)=ยฑโ144=ยฑ12โb =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,ย ยฑ6).**

**Ans.**We are given length of minor axis 16, foci (0,ยฑ6)

Length of minor axis, 2b = 16 โ b = 8

Foci, (0, ยฑc) = (0, ยฑ6)โ c = 6

It is known that a^{2}ย = b^{2ย }+ c^{2}.โa =ยฑโ(bยฒ+cยฒ) = ยฑโ(8ยฒ + 6ยฒ)= ยฑโ(64 + 36)=ยฑโ100=ยฑ10โa =10

The equation of the ellipse is

Putting the value a =10,b =8

โด The equation of the ellipse

**Q17.Foci (ยฑ3, 0), a = 4**

Ans. We are given that

Foci (ยฑ3, 0) and a = 4

Foci, ( ยฑc,0) = ( ยฑ3,0)โ c = 3

It is known that a^{2}ย = b^{2ย }+ c^{2}.โb =ยฑโ(aยฒ-cยฒ) = ยฑโ(4ยฒ – 3ยฒ)= ยฑโ(16- 9)=ยฑโ7=ยฑโ7โa =โ7

The equation of ellipse is

Putting b = โ7 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 =(ยฑc,0) = (ยฑ3,0)

It is known that aยฒ = bยฒ + cยฒ โ a =ยฑโ(bยฒ +cยฒ) = ยฑโ(3ยฒ +4ยฒ) =ยฑโ25=ยฑ5

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

bยฒ = 10 and aยฒ= 40.

Substituting the value of bยฒ = 10 and aยฒ= 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

bยฒ = 13 and aยฒ= 52.

Substituting the value of bยฒ = 13 and aยฒ= 52 in the equation,we get the required equation

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