Updated By SaraNextGen

On March 11, 2024, 11:35 AM

Question 1:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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)

Answer:

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 .

Question 8:

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

Answer:

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 .

Question 9:

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

Answer:

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 .

Question 10:

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

Answer:

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.

Answer:

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.

Answer:

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 .

Question 13:

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

Answer:

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 .

Question 14:

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

Answer:

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 .

Question 15:

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

Answer:

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 .