Hyperbola

Given Equation:

Comparing with the equation of hyperbola we get,

a = 3 and b = 4

(i) Length of Transverse axis = 2a = 6 units.

Length of Conjugate axis = 2b = 8 units.

(ii) Coordinates of the vertices = (a, 0) = (3, 0)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (ae, 0) = (5, 0)

(v) Length of the rectum = units.

Given Equation:

Comparing with the equation of hyperbola we get,

a = 5 and b = 2

(i) Length of Transverse axis = 2a = 10 units.

Length of Conjugate axis = 2b = 4 units.

(ii) Coordinates of the vertices = (a, 0) = (5, 0)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (ae, 0) = (, 0)

(v) Length of the rectum = units.

Given Equation: x² – y² = 1

Comparing with the equation of hyperbola we get,

a = 1 and b = 1

(i) Length of Transverse axis = 2a = 2 units.

Length of Conjugate axis = 2b = 2 units.

(ii) Coordinates of the vertices = (a, 0) = (1, 0)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (ae, 0) = (, 0)

(v) Length of the rectum = units.

Given Equation: 3x² – 2y² = 6 ⇒

Comparing with the equation of hyperbola we get,

a = and b =

(i) Length of Transverse axis = 2a = units.

Length of Conjugate axis = 2b = units.

(ii) Coordinates of the vertices = (a, 0) = (, 0)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (ae, 0) = (, 0)

(v) Length of the rectum = units.

Given Equation: 25x² – 9y² = 225 ⇒

Comparing with the equation of hyperbola we get,

a = 3 and b = 5

(i) Length of Transverse axis = 2a = 6 units.

Length of Conjugate axis = 2b = 10 units.

(ii) Coordinates of the vertices = (a, 0) = (3, 0)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (ae, 0) = (, 0)

(v) Length of the rectum = units.

Given Equation: 24x² – 25y² = 600 ⇒

Comparing with the equation of hyperbola we get,

a = 5 and b =

(i) Length of Transverse axis = 2a = 10 units.

Length of Conjugate axis = 2b = units.

(ii) Coordinates of the vertices = (a, 0) = (5, 0)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (ae, 0) = (7, 0)

(v) Length of the rectum = units.

Given Equation:

Comparing with the equation of hyperbola we get,

a = 4 and b = 7

(i) Length of Transverse axis = 2a = 8 units.

Length of Conjugate axis = 2b = 14 units.

(ii) Coordinates of the vertices = (0, a) = (0, 4)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (0, ae) = (0, )

(v) Length of the rectum = units.

Given Equation:

Comparing with the equation of hyperbola we get,

a = 3 and b =

(i) Length of Transverse axis = 2a = 6 units.

Length of Conjugate axis = 2b = units.

(ii) Coordinates of the vertices = (0, a) = (0, 3)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (0, ae) = (0, 6)

(v) Length of the rectum = units.

Given Equation: 3y² – x² = 108 ⇒

Comparing with the equation of hyperbola we get,

a = 6 and b =

(i) Length of Transverse axis = 2a = 12 units.

Length of Conjugate axis = 2b = units.

(ii) Coordinates of the vertices = (0, a) = (0, 6)

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (0, ae) = (0, 12)

(v) Length of the rectum = units.

Given Equation: 5y² – 9x² = 36 ⇒

Comparing with the equation of hyperbola we get,

a = and b = 2

(i) Length of Transverse axis = 2a = units.

Length of Conjugate axis = 2b = 4 units.

(ii) Coordinates of the vertices = (0, a) = (0, )

(iv) Here, eccentricity, e =

(iii) Coordinates of the foci = (0, ae) = (0, .) = (0, )

(v) Length of the rectum = units.

Given: Vertices at (±6, 0) and foci at (±8, 0)

Need to find: The equation of the hyperbola.

Let, the equation of the parabola be:

Vertices of the parabola is at (±6, 0)

That means a = 6

The foci are given at (±8, 0)

A and B are the foci.

That means, ae = 8, where e is the eccentricity.

⇒ [As a = 6]

⇒

We know that,

Therefore,

⇒

⇒ [Squaring both sides]

⇒

⇒

⇒ [As a = 6]

So, the equation of the hyperbola is,

⇒ [Answer]

Given: Vertices at (0, ±5) and foci at (0, ±8)

Need to find: The equation of the hyperbola.

Let, the equation of the parabola be:

Vertices of the parabola are at (0, ±5)

That means a = 5

The foci are given at (0, ±8)

A and B are the vertices. C and D are the foci.

That means, ae = 8, where e is the eccentricity.

⇒ [As a = 5]

⇒

We know that,

Therefore,

⇒

⇒ [Squaring both sides]

⇒

⇒

⇒ [As a = 5]

So, the equation of the hyperbola is,

⇒ [Answer]

Given: Foci are , the transverse axis is of the length 10

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

The transverse axis is of the length 10, i.e., 2a = 10

Therefore, a = 5

The foci are given at

A and B are the foci.

That means, , where e is the eccentricity.

⇒ [As a = 5]

⇒

We know that,

Therefore,

⇒

⇒ [Squaring both sides]

⇒

⇒

⇒ [As a = 5]

So, the equation of the hyperbola is,

⇒ [Answer]

Given: Foci are (±5, 0), the conjugate axis is of the length 8

Need to find: The equation of the hyperbola and eccentricity.

Let, the equation of the hyperbola be:

The conjugate axis is of the length 8, i.e., 2b = 8

Therefore, b = 4

The foci are given at (±5, 0)

A and B are the foci.

That means, ae = 5, where e is the eccentricity.

We know that,

Therefore,

⇒

⇒

⇒ [Squaring both sides]

⇒

⇒

⇒

So, the equation of the hyperbola is,

⇒

Eccentricity, [Answer]

Given: Foci are the length of the latus rectum is 8 units

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

The length of the latus rectum is 8 units.

Therefore, ⇒ ---- (1)

The foci are given at

That means, ae =, where e is the eccentricity.

We know that,

Therefore,

⇒

⇒

⇒ [Squaring both sides]

⇒ [From (1)]

⇒

⇒

⇒

So, either a = 5 or, a = -9

That means, either b = or, b =

The value of b = is not a valid one. So, the b value and its corresponding a value is not acceptable.

Hence, the acceptable value of a is 5 and b is

So, the equation of the hyperbola is,

⇒ [Answer]

Given: Vertices are (±2, 0) and the eccentricity is 2

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

Vertices are (±2, 0), that means, a = 2

And also given, the eccentricity, e = 2

A and B are the foci.

We know that,

Therefore,

⇒

⇒ [Squaring both sides]

⇒

⇒ [As a = 2]

So, the equation of the hyperbola is,

⇒ [Answer]

Given: Foci are , and the eccentricity is

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

The eccentricity, e =

And also given, foci are

That means, ae =

⇒

⇒ [As e =]

⇒

We know that,

Therefore,

⇒

⇒ [Squaring both sides]

⇒

⇒ [As a = ]

So, the equation of the hyperbola is,

⇒ [Answer]

Given: The length of latus rectum is 4, and the eccentricity is 3

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

The length of the latus rectum is 4 units.

Therefore, ⇒ ---- (1)

And also given, the eccentricity, e = 3

We know that,

Therefore,

⇒

⇒ [Squaring both sides]

⇒

⇒

⇒ [From (1)]

⇒

Therefore,

So, the equation of the hyperbola is,

⇒ ⇒ [Answer]

Given: Eccentricity is , and the distance between foci is 16

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

Distance between the foci is 16, i.e., 2ae = 16

And also given, the eccentricity, e =

Therefore,

---- (1)

We know that,

Therefore,

⇒

⇒ [Squaring both sides]

⇒

⇒ [From (1)]

So, the equation of the hyperbola is,

⇒ ⇒ [Answer]

Given: Vertices are (0, ±3) and the eccentricity is

Need to find: The equation of the hyperbola and coordinates of foci.

Let, the equation of the hyperbola be:

Vertices are (±3, 0), that means, a = 3

A and B are vertices.

And also given, the eccentricity, e =

We know that,

Therefore,

⇒

⇒ [Squaring both sides]

⇒

⇒ [As a = 3]

So, the equation of the hyperbola is,

⇒

Coordinates of the foci = (ae, 0) = (4, 0) [Answer]

Given: Foci are (0, ±13), the conjugate axis is of the length 24

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

The conjugate axis is of the length 24, i.e., 2b = 24

Therefore, b = 12

The foci are given at (0, ±13)

A and B are the foci.

That means, ae = 13, where e is the eccentricity.

We know that,

Therefore,

⇒

⇒

⇒ [Squaring both sides]

⇒ [As b = 12]

So, the equation of the hyperbola is,

⇒ [Answer]

Given: Foci are (0, ±10) and the length of latus rectum is 9 units

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

The length of the latus rectum is 9 units.

Therefore, ⇒ ---- (1)

The foci are given at (0, ±10)

That means, ae = 10, where e is the eccentricity.

We know that,

Therefore,

⇒

⇒

⇒ [Squaring both sides]

⇒ [From (1)]

⇒

⇒

⇒

So, either a = 16 or, a =

That means, either b = or, b =

The value of b = is not a valid one. So, the b value and its corresponding a value is not acceptable.

Hence, the acceptable value of a is 16 and b is

So, the equation of the hyperbola is,

⇒ [Answer]

Given: Foci at and passing through the point P(3, 4)

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

It passes through the point P(3, 4)

So putting the values of (x, y) we get,

⇒ ---- (1)

Foci at

So,

We know,

⇒

⇒

⇒ [Squaring on both sides]

⇒ ---- (2)

Comparing (1) and (2) we get,

Solving the equations we get,

b₁ =

b₂ =

b₃ =

b₄ =

With the help of any of these values of b we can’t find out the equation of the hyperbola.

* This is the only process we can apply in this standard.