** According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 10.*

**NCERT Solutions for Class 11 Maths Chapter 11 Exercise 11.4** have been carefully compiled and developed as per the latest update on the CBSE Syllabus. These solutions contain detailed step-by-step explanations of all the problems that are present in the NCERT textbook. Exercise 11.4 of NCERT Solutions for Class 11 Maths Chapter 11 â€“ Conic Sections is based on the following topics:

- Hyperbola
- Eccentricity
- Standard equation of Hyperbola
- Latus rectum

The solutions explain topics with engaging math activities that strengthen the concepts further. Each question of the exercise in NCERT Class 11 Maths Solutions has been carefully solved for the students to understand, keeping the board examinations in mind.

## NCERT Solutions for Class 11 Maths Chapter 11 â€“ Conic Sections Exercise 11.4

### Solutions for Class 11 Maths Chapter 11 â€“ Exercise 11.4

**In each of the exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.**

**1. x ^{2}/16 â€“ y^{2}/9 = 1**

**Solution:**

Given:

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

On comparing this equation with the standard equation of hyperbola x^{2}/a^{2} â€“ y^{2}/b^{2} = 1,

We get a = 4 and b = 3,

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

So,

c^{2}Â = 4^{2}Â + 3^{2}

=** âˆš**25

c = 5

Then,

The coordinates of the foci are (**Â±**5, 0).

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

Eccentricity, e = c/a = 5/4

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

**2. y ^{2}/9 â€“ x^{2}/27 = 1**

**Solution:**

Given:

The equation is y^{2}/9 â€“ x^{2}/27 = 1 or y^{2}/3^{2} â€“ x^{2}/27^{2} = 1

On comparing this equation with the standard equation of hyperbola y^{2}/a^{2} â€“ x^{2}/b^{2} = 1,

We get a = 3 and b = **âˆš**27,

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

So,

c^{2}Â = 3^{2}Â + (**âˆš**27)^{2}

= 9 + 27

c^{2} = 36

c = **âˆš**36

= 6

Then,

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

The coordinates of the vertices are (0, 3) and (0, â€“ 3).

Eccentricity, e = c/a = 6/3 = 2

Length of latus rectum = 2b^{2}/a = (2 Ã— 27)/3 = (54)/3 = 18

**3. 9y ^{2}Â â€“ 4x^{2}Â = 36**

**Solution:**

Given:

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

On comparing this equation with the standard equation of hyperbola y^{2}/a^{2} â€“ x^{2}/b^{2} = 1,

We get a = 2 and b = 3,

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

So,

c^{2}Â = 4Â + 9

c^{2} = 13

c = **âˆš**13

Then,

The coordinates of the foci are (0, **âˆš**13) and (0, â€“**âˆš**13).

The coordinates of the vertices are (0, 2) and (0, â€“ 2).

Eccentricity, e = c/a = **âˆš**13/2

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

**4. 16x ^{2}Â â€“ 9y^{2}Â = 576**

**Solution:**

Given:

The equation is 16x^{2}Â â€“ 9y^{2}Â = 576

Let us divide the whole equation by 576, we get

16x^{2}/576 â€“ 9y^{2}/576 = 576/576

x^{2}/36 â€“ y^{2}/64 = 1

On comparing this equation with the standard equation of hyperbola x^{2}/a^{2} â€“ y^{2}/b^{2} = 1,

We get a = 6 and b = 8,

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

So,

c^{2}Â = 36Â + 64

c^{2} = **âˆš**100

c = 10

Then,

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

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

Eccentricity, e = c/a = 10/6 = 5/3

Length of latus rectum = 2b^{2}/a = (2 Ã— 8^{2})/6 = (2Ã—64)/6 = 64/3

**5. 5y ^{2}Â â€“ 9x^{2}Â = 36**

**Solution:**

Given:

The equation is 5y^{2}Â â€“ 9x^{2}Â = 36

Let us divide the whole equation by 36. We get

5y^{2}/36 â€“ 9x^{2}/36 = 36/36

y^{2}/(36/5) â€“ x^{2}/4 = 1

On comparing this equation with the standard equation of hyperbola y^{2}/a^{2} â€“ x^{2}/b^{2} = 1,

We get a = 6/**âˆš**5 and b = 2,

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

So,

c^{2}Â = 36/5Â + 4

c^{2} = 56/5

c = **âˆš**(56/5)

= 2**âˆš**14/**âˆš**5

Then,

The coordinates of the foci are (0, 2**âˆš**14/**âˆš**5) and (0, â€“ 2**âˆš**14/**âˆš**5).

The coordinates of the vertices are (0, 6/**âˆš**5) and (0, -6/**âˆš**5).

Eccentricity, e = c/a = (2**âˆš**14/**âˆš**5) / (6/**âˆš**5) = **âˆš**14/3

Length of latus rectum = 2b^{2}/a = (2 Ã— 2^{2})/6/âˆš5 = (2Ã—4)/6/**âˆš**5 = 4**âˆš**5/3

**6. 49y ^{2}Â â€“ 16x^{2}Â = 784.**

**Solution:**

Given:

The equation is 49y^{2}Â â€“ 16x^{2}Â = 784.

Let us divide the whole equation by 784. We get

49y^{2}/784 â€“ 16x^{2}/784 = 784/784

y^{2}/16 â€“ x^{2}/49 = 1

^{2}/a^{2} â€“ x^{2}/b^{2} = 1,

We get a = 4 and b = 7,

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

So,

c^{2}Â = 16Â + 49

c^{2} = 65

c = **âˆš**65

Then,

The coordinates of the foci are (0, **âˆš**65) and (0, â€“**âˆš**65).

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

Eccentricity, e = c/a = **âˆš**65/4

Length of latus rectum = 2b^{2}/a = (2 Ã— 7^{2})/4 = (2Ã—49)/4 = 49/2

**In each Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions**

**7. Vertices (Â±2, 0), foci (Â±3, 0)**

**Solution:**

Given:

Vertices (Â±2, 0) and foci (Â±3, 0)

Here, the vertices are on the x-axis.

So, the equation of the hyperbola is of the form x^{2}/a^{2} â€“ y^{2}/b^{2} = 1

Since, the vertices are (Â±2, 0), so, a = 2

Since, the foci are (Â±3, 0), so, c = 3

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

So, 2^{2}Â + b^{2}Â = 3^{2}

b^{2}Â = 9 â€“ 4 = 5

âˆ´ The equation of the hyperbola is x^{2}/4 â€“ y^{2}/5 = 1

**8. Vertices (0,Â Â±Â 5), foci (0,Â Â±Â 8)**

**Solution:**

Given:

Vertices (0,Â Â±Â 5) and foci (0,Â Â±Â 8)

Here, the vertices are on the y-axis.

So, the equation of the hyperbola is of the form y^{2}/a^{2} â€“ x^{2}/b^{2} = 1

Since the vertices are (0, Â±5), a = 5

Since the foci are (0, Â±8), c = 8

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

So, 5^{2}Â + b^{2}Â = 8^{2}

b^{2}Â = 64 â€“ 25 = 39

âˆ´ The equation of the hyperbola is y^{2}/25 â€“ x^{2}/39 = 1

**9. Vertices (0,Â Â±Â 3), foci (0,Â Â±Â 5)**

**Solution:**

Given:

Vertices (0,Â Â±Â 3) and foci (0,Â Â±Â 5)

Here, the vertices are on the y-axis.

So, the equation of the hyperbola is of the formÂ y^{2}/a^{2} â€“ x^{2}/b^{2} = 1

Since the vertices are (0, Â±3), so, a = 3

Since the foci are (0, Â±5), so, c = 5

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

So, 3^{2}Â + b^{2}Â = 5^{2}

b^{2}Â = 25 â€“ 9 = 16

âˆ´ The equation of the hyperbola is y^{2}/9 â€“ x^{2}/16 = 1

**10. Foci (Â±5, 0), the transverse axis is of length 8.**

**Solution:**

Given:

Foci (Â±5, 0) and the transverse axis is of length 8.

Here, the foci are on the x-axis.

The equation of the hyperbola is of the form x^{2}/a^{2} â€“ y^{2}/b^{2} = 1

Since, the foci are (Â±5, 0), so, c = 5

Since the length of the transverse axis is 8,

2a = 8

a = 8/2

= 4

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

4^{2}Â + b^{2}Â = 5^{2}

b^{2}Â = 25 â€“ 16

= 9

âˆ´ The equation of the hyperbola is x^{2}/16 â€“ y^{2}/9 = 1

**11. Foci (0,Â Â±13), the conjugate axis is of length 24.**

**Solution:**

Given:

Foci (0,Â Â±13) and the conjugate axis is of length 24.

Here, the foci are on the y-axis.

The equation of the hyperbola is of the formÂ y^{2}/a^{2} â€“ x^{2}/b^{2} = 1

Since the foci are (0, Â±13), so, c = 13

Since the length of the conjugate axis is 24,

2b = 24

b = 24/2

= 12

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

a^{2}Â + 12^{2}Â = 13^{2}

a^{2}Â = 169 â€“ 144

= 25

âˆ´ The equation of the hyperbola is y^{2}/25 â€“ x^{2}/144 = 1

**12. Foci (Â±Â 3âˆš5, 0), the latus rectum is of length 8.**

**Solution:**

Given:

Foci (Â±Â 3âˆš5, 0) and the latus rectum is of length 8.

Here, the foci are on the x-axis.

The equation of the hyperbola is of the formÂ x^{2}/a^{2} â€“ y^{2}/b^{2} = 1

Since, the foci are (Â±Â 3âˆš5, 0), so, c =Â Â±Â 3âˆš5

Length of the latus rectum is 8

2b^{2}/a = 8

2b^{2}Â = 8a

b^{2} = 8a/2

= 4a

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

a^{2}Â + 4a = 45

a^{2}Â + 4a â€“ 45 = 0

a^{2}Â + 9a â€“ 5a â€“ 45 = 0

(a + 9) (a -5) = 0

a = -9 or 5

Since a is nonâ€“negative, a = 5

So, b^{2}Â = 4a

= 4 Ã— 5

= 20

âˆ´ The equation of the hyperbola is x^{2}/25 â€“ y^{2}/20 = 1

**13. Foci (Â±Â 4, 0), the latus rectum is of length 12**

**Solution:**

Given:

Foci (Â±Â 4, 0) and the latus rectum is of length 12

Here, the foci are on the x-axis.

The equation of the hyperbola is of form x^{2}/a^{2} â€“ y^{2}/b^{2} = 1

Since, the foci areÂ (Â±Â 4, 0),Â so, c = 4

Length of the latus rectum is 12

2b^{2}/a = 12

2b^{2}Â = 12a

b^{2} = 12a/2

= 6a

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

a^{2}Â + 6a = 16

a^{2}Â + 6a â€“ 16 = 0

a^{2}Â + 8a â€“ 2a â€“ 16 = 0

(a + 8) (a â€“ 2) = 0

a = -8 or 2

Since a is nonâ€“negative, a = 2

So, b^{2}Â = 6a

= 6 Ã— 2

= 12

âˆ´ The equation of the hyperbola isÂ x^{2}/4 â€“ y^{2}/12 = 1

**14. Vertices (Â±7, 0), e = 4/3**

**Solution:**

Given:

Vertices (Â±7, 0) and e = 4/3

Here, the vertices are on the x-axis

The equation of the hyperbola is of form x^{2}/a^{2} â€“ y^{2}/b^{2} = 1

Since, theÂ vertices are (Â±Â 7, 0), so, a = 7

It is given that e =Â 4/3

c/a = 4/3

3c = 4a

Substitute the value of a, and we get

3c = 4(7)

c = 28/3

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

7^{2}Â + b^{2}Â = (28/3)^{2}

b^{2} = 784/9 â€“ 49

= (784 â€“ 441)/9

= 343/9

âˆ´ The equation of the hyperbola is x^{2}/49 â€“ 9y^{2}/343 = 1

**15. Foci (0,Â Â±âˆš10), passing through (2, 3)**

**Solution:**

Given:

Foci (0,Â Â±âˆš10) and passing through (2, 3)

Here, the foci are on the y-axis.

The equation of the hyperbola is of the form y^{2}/a^{2} â€“ x^{2}/b^{2} = 1

Since, the foci areÂ (Â±âˆš10, 0),Â so, c =Â âˆš10

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

b^{2}Â = 10 â€“ a^{2}Â â€¦â€¦â€¦â€¦.. (1)

It is given that the hyperbola passes through points (2, 3)

So, 9/a^{2} â€“ 4/b^{2} = 1 â€¦ (2)

From equations (1) and (2), we get,

9/a^{2} â€“ 4/(10-a^{2}) = 1

9(10 â€“ a^{2}) â€“ 4a^{2}Â = a^{2}(10 â€“a^{2})

90 â€“ 9a^{2}Â â€“ 4a^{2}Â = 10a^{2}Â â€“ a^{4}

a^{4}Â â€“ 23a^{2}Â + 90 = 0

a^{4}Â â€“ 18a^{2}Â â€“ 5a^{2 }+ 90 = 0

a^{2}(a^{2}Â -18) -5(a^{2}Â -18) = 0

(a^{2}Â â€“ 18) (a^{2}Â -5) = 0

a^{2}Â = 18 or 5

In hyperbola, c > a i.e., c^{2 }> a^{2}

So, a^{2}Â = 5

b^{2}Â = 10 â€“ a^{2}

= 10 â€“ 5

= 5

âˆ´ The equation of the hyperbola is y^{2}/5 â€“ x^{2}/5 = 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 11.3 Solutions 20 Questions

Miscellaneous Exercise On Chapter 11 Solutions 8 Questions

Also explore â€“ NCERT Class 11 Solutions

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