NCERT Solutions: Exercise 10.4 - Conic Sections

Q1: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 
Ans: The given equation is
On comparing this equation with the standard equation of hyperbola i.e., , we obtain a = 4 and b = 3.
We know that a² +  b² = c².

 
Therefore,
The coordinates of the foci are (±5, 0).
The coordinates of the vertices are (±4, 0).

 
Length of latus rectum

Q2: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola
 
Ans: The given equation is .
On comparing this equation with the standard equation of hyperbola i.e., , we obtain a = 3 and .
We know that a² +  b² = c².

 
Therefore,
The coordinates of the foci are (0, ±6).
The coordinates of the vertices are (0, ±3).

 
Length of latus rectum

Q3: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 9y² - 4x² = 36
Ans: The given equation is 9y² - 4x² = 36.
It can be written as
9y² - 4x² = 36

 
On comparing equation (1) with the standard equation of hyperbola i.e., , we obtain a = 2 and b = 3.
We know that a² +  b² = c².

 
Therefore,
The coordinates of the foci are .
The coordinates of the vertices are .

 
Length of latus rectum

Q4: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 16x² - 9y² = 576
Ans: The given equation is 16x² - 9y² = 576.
It can be written as
16x² - 9y² = 576

 
On comparing equation (1) with the standard equation of hyperbola i.e., , we obtain a = 6 and b = 8.
We know that a²  + b² = c².

 
Therefore,
The coordinates of the foci are (±10, 0).
The coordinates of the vertices are (±6, 0).

 
Length of latus rectum

Q5: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 5y² - 9x² = 36
Ans: The given equation is 5y² - 9x² = 36.

 
On comparing equation (1) with the standard equation of hyperbola i.e., , we obtain a =   and b = 2.
We know that a²  + b² = c².

 
Therefore, the coordinates of the foci are .
The coordinates of the vertices are .

 
Length of latus rectum

Q6: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 49y² - 16x² = 784
Ans: The given equation is 49y² - 16x² = 784.
It can be written as
49y² - 16x² = 784

 
On comparing equation (1) with the standard equation of hyperbola i.e., , we obtain a = 4 and b = 7.
We know that a²  + b² = c².

 
Therefore,
The coordinates of the foci are .
The coordinates of the vertices are (0, ±4).

 
Length of latus rectum

Q7: Find the equation of the hyperbola satisfying the give conditions: Vertices (±2, 0), foci (±3, 0)
Ans: Vertices (±2, 0), foci (±3, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the hyperbola is of the form .
Since the vertices are (±2, 0), a = 2.
Since the foci are (±3, 0), c = 3.
We know that a²  + b² = c².

 
Thus, the equation of the hyperbola is .

Q8: Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±5), foci (0, ±8)
Ans: Vertices (0, ±5), foci (0, ±8)
Here, the vertices are on the y-axis.
Therefore, the equation of the hyperbola is of the form
Since the vertices are (0, ±5), a = 5.
Since the foci are (0, ±8), c = 8.
We know that a²  + b² = c².

 
Thus, the equation of the hyperbola is .

Q9: Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±3), foci (0, ±5)
Ans: Vertices (0, ±3), foci (0, ±5)
Here, the vertices are on the y-axis.
Therefore, the equation of the hyperbola is of the form .
Since the vertices are (0, ±3), a = 3.
Since the foci are (0, ±5), c = 5.
We know that a²  + b² = c².
∴ 3² +  b² = 5²
⇒ b² = 25 - 9 = 16
Thus, the equation of the hyperbola is .

Q10: Find the equation of the hyperbola satisfying the give conditions: Foci (±5, 0), the transverse axis is of length 8.
Ans: Foci (±5, 0), the transverse axis is of length 8.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form .
Since the foci are (±5, 0), c = 5.
Since the length of the transverse axis is 8, 2a = 8 ⇒ a = 4.
We know that a² +  b² = c².
∴ 4² +  b² = 5²
⇒ b² = 25 - 16 = 9
Thus, the equation of the hyperbola is .

Q11: Find the equation of the hyperbola satisfying the give conditions: Foci (0, ±13), the conjugate axis is of length 24.
Ans: Foci (0, ±13), the conjugate axis is of length 24.
Here, the foci are on the y-axis.
Therefore, the equation of the hyperbola is of the form .
Since the foci are (0, ±13), c = 13.
Since the length of the conjugate axis is 24, 2b = 24 ⇒ b = 12.
We know that a²  + b² = c².
∴ a² + 12² = 13²
⇒ a² = 169 - 144 = 25
Thus, the equation of the hyperbola is .

Q12: Find the equation of the hyperbola satisfying the give conditions: Foci , the latus rectum is of length 8.
Ans: Foci , the latus rectum is of length 8.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form .
Since the foci are , c = .
Length of latus rectum = 8

We know that a²  + b² = c².
∴ a² + 4a = 45
⇒ a² + 4a - 45 = 0
⇒ a² + 9a - 5a - 45 = 0
⇒ (a + 9) (a - 5) = 0
⇒ a = -9, 5
Since a is non-negative, a = 5.
∴ b² = 4a = 4 × 5 = 20
Thus, the equation of the hyperbola is .

Q13: Find the equation of the hyperbola satisfying the give conditions: Foci (±4, 0), the latus rectum is of length 12
Ans: Foci (±4, 0), the latus rectum is of length 12.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form .
Since the foci are (±4, 0), c = 4.
Length of latus rectum = 12

 
We know that a²  + b² = c².
∴ a² + 6a = 16
⇒ a² + 6a - 16 = 0
⇒ a² + 8a - 2a - 16 = 0
⇒ (a  8) (a - 2) = 0
⇒ a = -8, 2
Since a is non-negative, a = 2.
∴ b² = 6a = 6 × 2 = 12
Thus, the equation of the hyperbola is  .

Q14: Find the equation of the hyperbola satisfying the give conditions: Vertices (±7, 0), 
Ans: Vertices (±7, 0),
Here, the vertices are on the x-axis.
Therefore, the equation of the hyperbola is of the form .
Since the vertices are (±7, 0), a = 7.
It is given that

We know that a² +  b² = c².

 
Thus, the equation of the hyperbola is .

Q15: Find the equation of the hyperbola satisfying the give conditions: Foci , passing through (2, 3)
Ans: Foci , passing through (2, 3)
Here, the foci are on the y-axis.
Therefore, the equation of the hyperbola is of the form .
Since the foci are , c = .
We know that a² +  b² = c².
∴ a² +  b² = 10
⇒ b² = 10 - a² ... (1)
Since the hyperbola passes through point (2, 3),

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

 
In hyperbola, c > a, i.e., c² > a²
∴ a² = 5
⇒ b² = 10 - a² = 10 - 5 = 5
Thus, the equation of the hyperbola is .