## NCERT Solutions for Class 11 Maths Chapter 11 – Conic Sections Ex 11.4

#### Question 1:

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola

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

#### Question 2:

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola

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

#### Question 3:

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 9y2 – 4x2 = 36

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

#### Question 4:

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 16x2 – 9y2 = 576

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

#### Question 5:

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 5y2 – 9x2 = 36

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

#### Question 6:

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 49y2 – 16x2 = 784

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

#### Question 7:

Find the equation of the hyperbola satisfying the give conditions: Vertices (±2, 0), foci (±3, 0)

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), = 2.
Since the foci are (±3, 0), c = 3.
We know that a2 + b2 = c2.
Thus, the equation of the hyperbola is.

#### Question 8:

Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±5), foci (0, ±8)

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), = 5.
Since the foci are (0, ±8), c = 8.
We know that a2 + b2 = c2.
Thus, the equation of the hyperbola is.

#### Question 9:

Find the equation of the hyperbola satisfying the give conditions: Vertices (0, ±3), foci (0, ±5)

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), = 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.

#### Question 10:

Find the equation of the hyperbola satisfying the give conditions: Foci (±5, 0), the transverse axis is of length 8.

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.

#### Question 11:

Find the equation of the hyperbola satisfying the give conditions: Foci (0, ±13), the conjugate axis is of length 24.

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.

#### Question 12:

Find the equation of the hyperbola satisfying the give conditions: Foci, the latus rectum is of length 8.

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 arec =.
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, = 5.
b2 = 4= 4 × 5 = 20
Thus, the equation of the hyperbola is.

#### Question 13:

Find the equation of the hyperbola satisfying the give conditions: Foci (±4, 0), the latus rectum is of length 12

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, = 2.
b2 = 6= 6 × 2 = 12
Thus, the equation of the hyperbola is.

#### Question 14:

Find the equation of the hyperbola satisfying the give conditions: Vertices (±7, 0),

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), = 7.
It is given that
We know that a2 + b2 = c2.
Thus, the equation of the hyperbola is.

#### Question 15:

Find the equation of the hyperbola satisfying the give conditions: Foci, passing through (2, 3)

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 arec =.
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.

Courtesy : CBSE