Chapter 11 Conic Sections Ex 11.4 Solutions
Question - 1 : - Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 
Answer - 1 : -
The given equation is
.
On comparing this equation with the standardequation of hyperbola i.e.,
, we obtain a =4 and b = 3.
We know that a2 + b2 = c2.
Therefore,
The coordinates of thefoci are (±5, 0).
The coordinates of thevertices are (±4, 0).
Lengthof 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 - 2 : -
The given equation is
.
On comparing this equation with the standardequation of hyperbola i.e., 
, we obtain a =3 and
.
We know that a2 + b2 = c2.
Therefore,
The coordinates of thefoci are (0, ±6).
The coordinates of thevertices are (0, ±3).
Lengthof latus rectum
Question - 3 : - Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola9y2 – 4x2 = 36
Answer - 3 : -
The given equation is 9y2 – 4x2 = 36.
It can be written as
9y2 – 4x2 = 36
On comparing equation (1) with the standardequation of hyperbola i.e.,, we obtain a =2 and b = 3.
We know that a2 + b2 = c2.
Therefore,
The coordinates of thefoci are
.
The coordinates of thevertices are
.
Lengthof latus rectum
Question - 4 : - Find the coordinates of the foci and thevertices, the eccentricity, and the length of the latus rectum of the hyperbola16x2 – 9y2 = 576
Answer - 4 : -
The given equation is 16x2 – 9y2 = 576.
It can be written as
16x2 – 9y2 = 576
On comparing equation (1) with the standardequation of hyperbola i.e.,, we obtain a =6 and b = 8.
We know that a2 + b2 = c2.
Therefore,
The coordinates of thefoci are (±10, 0).
The coordinates of thevertices are (±6, 0).
Lengthof latus rectum
Question - 5 : - Find the coordinates of the foci and thevertices, the eccentricity, and the length of the latus rectum of the hyperbola5y2 – 9x2 = 36
Answer - 5 : -
The given equation is 5y2 – 9x2 = 36.
On comparing equation (1) with the standardequation of hyperbola i.e.,, we obtain a = 
and b = 2.
We know that a2 + b2 = c2.
Therefore, the coordinatesof the foci are
.
The coordinates of thevertices are
.
Lengthof latus rectum
Question - 6 : - Find the coordinates of the foci and thevertices, the eccentricity, and the length of the latus rectum of the hyperbola49y2 – 16x2 = 784
Answer - 6 : -
The given equation is 49y2 – 16x2 = 784.
It can be written as 49y2 – 16x2 = 784
On comparing equation (1) with the standardequation of hyperbola i.e.,, we obtain a =4 and b = 7.
We know that a2 + b2 = c2.
Therefore,
The coordinates of thefoci are
.
The coordinates of thevertices 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 - 7 : -
Vertices (±2, 0), foci(±3, 0)
Here, the vertices are on the x-axis.
Therefore, the equation ofthe 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 - 8 : -
Vertices (0, ±5), foci (0,±8)
Here, the vertices are on the y-axis.
Therefore, the equation ofthe 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 - 9 : -
Vertices (0, ±3), foci (0,±5)
Here, the vertices are on the y-axis.
Therefore, the equation ofthe 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 - 10 : -
Foci (±5, 0), thetransverse axis is of length 8.
Here, the foci are on the x-axis.
Therefore, the equation ofthe hyperbola is of the form.
Since the foci are (±5, 0), c =5.
Since the length of the transverse axis is8, 2a = 8 ⇒ a = 4.
We know that a2 + b2 = c2.
∴42 + b2 = 52
⇒ b2 = 25 – 16 = 9
Thus,the equation of the hyperbola is
.