RD Sharma Class 11 Solutions Chapter - Hyperbola

 Page 1


27. Hyperbola
Exercise 27.1
1. Question
The equation of the directrix of a hyperbola is x – y + 3 = 0. Its focus is (-1, 1) and eccentricity 3. Find the
equation of the hyperbola.
Answer
Given: Equation of directrix of a hyperbola is x – y + 3 = 0. Focus of hyperbola is (-1, 1) and eccentricity (e)
= 3
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + 1 + 2x + y
2
 + 1 – 2y} = 9{x
2
 + y
2
+ 9 + 6x – 6y – 2xy}
? 2x
2
 + 2 + 4x + 2y
2
 + 2 – 4y = 9x
2
 + 9y
2
+ 81 + 54x – 54y – 18xy
? 2x
2
 + 4 + 4x + 2y
2
– 4y – 9x
2
 - 9y
2
 - 81 – 54x + 54y + 18xy = 0
? – 7x
2
 - 7y
2
 – 50x + 50y + 18xy – 77 = 0
? 7x
2
 + 7y
2
 + 50x – 50y – 18xy + 77 = 0
This is the required equation of hyperbola
Page 2


27. Hyperbola
Exercise 27.1
1. Question
The equation of the directrix of a hyperbola is x – y + 3 = 0. Its focus is (-1, 1) and eccentricity 3. Find the
equation of the hyperbola.
Answer
Given: Equation of directrix of a hyperbola is x – y + 3 = 0. Focus of hyperbola is (-1, 1) and eccentricity (e)
= 3
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + 1 + 2x + y
2
 + 1 – 2y} = 9{x
2
 + y
2
+ 9 + 6x – 6y – 2xy}
? 2x
2
 + 2 + 4x + 2y
2
 + 2 – 4y = 9x
2
 + 9y
2
+ 81 + 54x – 54y – 18xy
? 2x
2
 + 4 + 4x + 2y
2
– 4y – 9x
2
 - 9y
2
 - 81 – 54x + 54y + 18xy = 0
? – 7x
2
 - 7y
2
 – 50x + 50y + 18xy – 77 = 0
? 7x
2
 + 7y
2
 + 50x – 50y – 18xy + 77 = 0
This is the required equation of hyperbola
2 A. Question
Find the equation of the hyperbola whose
focus is (0, 3), directrix is x + y – 1 = 0 and eccentricity = 2
Answer
Given: Equation of directrix of a hyperbola is x + y – 1 = 0. Focus of hyperbola is (0, 3) and eccentricity (e) =
2
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + y
2
 + 9 – 6y} = 4{x
2
 + y
2
 + 1 – 2x – 2y + 2xy}
Page 3


27. Hyperbola
Exercise 27.1
1. Question
The equation of the directrix of a hyperbola is x – y + 3 = 0. Its focus is (-1, 1) and eccentricity 3. Find the
equation of the hyperbola.
Answer
Given: Equation of directrix of a hyperbola is x – y + 3 = 0. Focus of hyperbola is (-1, 1) and eccentricity (e)
= 3
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + 1 + 2x + y
2
 + 1 – 2y} = 9{x
2
 + y
2
+ 9 + 6x – 6y – 2xy}
? 2x
2
 + 2 + 4x + 2y
2
 + 2 – 4y = 9x
2
 + 9y
2
+ 81 + 54x – 54y – 18xy
? 2x
2
 + 4 + 4x + 2y
2
– 4y – 9x
2
 - 9y
2
 - 81 – 54x + 54y + 18xy = 0
? – 7x
2
 - 7y
2
 – 50x + 50y + 18xy – 77 = 0
? 7x
2
 + 7y
2
 + 50x – 50y – 18xy + 77 = 0
This is the required equation of hyperbola
2 A. Question
Find the equation of the hyperbola whose
focus is (0, 3), directrix is x + y – 1 = 0 and eccentricity = 2
Answer
Given: Equation of directrix of a hyperbola is x + y – 1 = 0. Focus of hyperbola is (0, 3) and eccentricity (e) =
2
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + y
2
 + 9 – 6y} = 4{x
2
 + y
2
 + 1 – 2x – 2y + 2xy}
? 2x
2
 + 2y
2
 + 18 – 12y = 4x
2
 + 4y
2
+ 4 – 8x – 8y + 8xy
? 2x
2
 + 2y
2
 + 18 – 12y – 4x
2
 – 4y
2
 – 4 – 8x + 8y – 8xy = 0
? – 2x
2
 – 2y
2
 – 8x – 4y – 8xy + 14 = 0
? -2(x
2
 + y
2
 – 4x + 2y + 4xy – 7) = 0
? x
2
 + y
2
 – 4x + 2y + 4xy – 7 = 0
This is the required equation of hyperbola
2 B. Question
Find the equation of the hyperbola whose
focus is (1, 1), directrix is 3x + 4y + 8 = 0 and eccentricity = 2
Answer
Given: Equation of directrix of a hyperbola is 3x + 4y + 8 = 0. Focus of hyperbola is (1, 1) and eccentricity
(e) = 2
To find: equation of hyperbola
Let M be the point on directrix and P(x, y) be any point of hyperbola
Formula used:
where e is eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
Page 4


27. Hyperbola
Exercise 27.1
1. Question
The equation of the directrix of a hyperbola is x – y + 3 = 0. Its focus is (-1, 1) and eccentricity 3. Find the
equation of the hyperbola.
Answer
Given: Equation of directrix of a hyperbola is x – y + 3 = 0. Focus of hyperbola is (-1, 1) and eccentricity (e)
= 3
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + 1 + 2x + y
2
 + 1 – 2y} = 9{x
2
 + y
2
+ 9 + 6x – 6y – 2xy}
? 2x
2
 + 2 + 4x + 2y
2
 + 2 – 4y = 9x
2
 + 9y
2
+ 81 + 54x – 54y – 18xy
? 2x
2
 + 4 + 4x + 2y
2
– 4y – 9x
2
 - 9y
2
 - 81 – 54x + 54y + 18xy = 0
? – 7x
2
 - 7y
2
 – 50x + 50y + 18xy – 77 = 0
? 7x
2
 + 7y
2
 + 50x – 50y – 18xy + 77 = 0
This is the required equation of hyperbola
2 A. Question
Find the equation of the hyperbola whose
focus is (0, 3), directrix is x + y – 1 = 0 and eccentricity = 2
Answer
Given: Equation of directrix of a hyperbola is x + y – 1 = 0. Focus of hyperbola is (0, 3) and eccentricity (e) =
2
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + y
2
 + 9 – 6y} = 4{x
2
 + y
2
 + 1 – 2x – 2y + 2xy}
? 2x
2
 + 2y
2
 + 18 – 12y = 4x
2
 + 4y
2
+ 4 – 8x – 8y + 8xy
? 2x
2
 + 2y
2
 + 18 – 12y – 4x
2
 – 4y
2
 – 4 – 8x + 8y – 8xy = 0
? – 2x
2
 – 2y
2
 – 8x – 4y – 8xy + 14 = 0
? -2(x
2
 + y
2
 – 4x + 2y + 4xy – 7) = 0
? x
2
 + y
2
 – 4x + 2y + 4xy – 7 = 0
This is the required equation of hyperbola
2 B. Question
Find the equation of the hyperbola whose
focus is (1, 1), directrix is 3x + 4y + 8 = 0 and eccentricity = 2
Answer
Given: Equation of directrix of a hyperbola is 3x + 4y + 8 = 0. Focus of hyperbola is (1, 1) and eccentricity
(e) = 2
To find: equation of hyperbola
Let M be the point on directrix and P(x, y) be any point of hyperbola
Formula used:
where e is eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 25{x
2
 + 1 – 2x + y
2
 + 1 – 2y} = 4{9x
2
 + 16y
2
+ 64 + 24xy + 64y + 48x}
? 25x
2
 + 25 – 50x + 25y
2
 + 25 – 50y = 36x
2
 + 64y
2
 + 256 + 96xy + 256y + 192x
? 25x
2
 + 25 – 50x + 25y
2
 + 25 – 50y – 36x
2
 – 64y
2
 – 256 – 96xy – 256y – 192x = 0
? – 11x
2
 – 39y
2
 – 242x – 306y – 96xy – 206 = 0
? 11x
2
 + 39y
2
 + 242x + 306y + 96xy + 206 = 0
This is the required equation of hyperbola
2 C. Question
Find the equation of the hyperbola whose
focus is (1, 1) directrix is 2x + y = 1 and eccentricity = 
Answer
Given: Equation of directrix of a hyperbola is 2x + y – 1 = 0. Focus of hyperbola is (1, 1) and eccentricity (e)
= 
To find: equation of hyperbola
Let M be the point on directrix and P(x, y) be any point of hyperbola
Formula used:
where e is eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Page 5


27. Hyperbola
Exercise 27.1
1. Question
The equation of the directrix of a hyperbola is x – y + 3 = 0. Its focus is (-1, 1) and eccentricity 3. Find the
equation of the hyperbola.
Answer
Given: Equation of directrix of a hyperbola is x – y + 3 = 0. Focus of hyperbola is (-1, 1) and eccentricity (e)
= 3
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + 1 + 2x + y
2
 + 1 – 2y} = 9{x
2
 + y
2
+ 9 + 6x – 6y – 2xy}
? 2x
2
 + 2 + 4x + 2y
2
 + 2 – 4y = 9x
2
 + 9y
2
+ 81 + 54x – 54y – 18xy
? 2x
2
 + 4 + 4x + 2y
2
– 4y – 9x
2
 - 9y
2
 - 81 – 54x + 54y + 18xy = 0
? – 7x
2
 - 7y
2
 – 50x + 50y + 18xy – 77 = 0
? 7x
2
 + 7y
2
 + 50x – 50y – 18xy + 77 = 0
This is the required equation of hyperbola
2 A. Question
Find the equation of the hyperbola whose
focus is (0, 3), directrix is x + y – 1 = 0 and eccentricity = 2
Answer
Given: Equation of directrix of a hyperbola is x + y – 1 = 0. Focus of hyperbola is (0, 3) and eccentricity (e) =
2
To find: equation of the hyperbola
Let M be the point on directrix and P(x, y) be any point of the hyperbola
Formula used:
where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 2{x
2
 + y
2
 + 9 – 6y} = 4{x
2
 + y
2
 + 1 – 2x – 2y + 2xy}
? 2x
2
 + 2y
2
 + 18 – 12y = 4x
2
 + 4y
2
+ 4 – 8x – 8y + 8xy
? 2x
2
 + 2y
2
 + 18 – 12y – 4x
2
 – 4y
2
 – 4 – 8x + 8y – 8xy = 0
? – 2x
2
 – 2y
2
 – 8x – 4y – 8xy + 14 = 0
? -2(x
2
 + y
2
 – 4x + 2y + 4xy – 7) = 0
? x
2
 + y
2
 – 4x + 2y + 4xy – 7 = 0
This is the required equation of hyperbola
2 B. Question
Find the equation of the hyperbola whose
focus is (1, 1), directrix is 3x + 4y + 8 = 0 and eccentricity = 2
Answer
Given: Equation of directrix of a hyperbola is 3x + 4y + 8 = 0. Focus of hyperbola is (1, 1) and eccentricity
(e) = 2
To find: equation of hyperbola
Let M be the point on directrix and P(x, y) be any point of hyperbola
Formula used:
where e is eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 25{x
2
 + 1 – 2x + y
2
 + 1 – 2y} = 4{9x
2
 + 16y
2
+ 64 + 24xy + 64y + 48x}
? 25x
2
 + 25 – 50x + 25y
2
 + 25 – 50y = 36x
2
 + 64y
2
 + 256 + 96xy + 256y + 192x
? 25x
2
 + 25 – 50x + 25y
2
 + 25 – 50y – 36x
2
 – 64y
2
 – 256 – 96xy – 256y – 192x = 0
? – 11x
2
 – 39y
2
 – 242x – 306y – 96xy – 206 = 0
? 11x
2
 + 39y
2
 + 242x + 306y + 96xy + 206 = 0
This is the required equation of hyperbola
2 C. Question
Find the equation of the hyperbola whose
focus is (1, 1) directrix is 2x + y = 1 and eccentricity = 
Answer
Given: Equation of directrix of a hyperbola is 2x + y – 1 = 0. Focus of hyperbola is (1, 1) and eccentricity (e)
= 
To find: equation of hyperbola
Let M be the point on directrix and P(x, y) be any point of hyperbola
Formula used:
where e is eccentricity, PM is perpendicular from any point P on hyperbola to the directrix
Therefore,
Squaring both sides:
{? (a – b)
2
 = a
2
 + b
2
 + 2ab &
(a + b + c)
2
 = a
2
 + b
2
 + c
2
 + 2ab + 2bc + 2ac}
? 5{x
2
 + 1 – 2x + y
2
 + 1 – 2y} = 3{4x
2
 + y
2
+ 1 + 4xy – 2y – 4x}
? 5x
2
 + 5 – 10x + 5y
2
 + 5 – 10y = 12x
2
 + 3y
2
 + 3 + 12xy – 6y – 12x
? 5x
2
 + 5 – 10x + 5y
2
 + 5 – 10y – 12x
2
 – 3y
2
 – 3 – 12xy + 6y + 12x = 0
? – 7x
2
 + 2y
2
 + 2x – 4y – 12xy + 7 = 0
? 7x
2
 – 2y
2
 – 2x + 4y + 12xy – 7 = 0
This is the required equation of hyperbola.
2 D. Question
Find the equation of the hyperbola whose
focus is (2, -1), directrix is 2x + 3y = 1 and eccentricity = 2
Answer
Given: Equation of directrix of a hyperbola is 2x + 3y – 1 = 0. Focus of hyperbola is (2, -1) and eccentricity
(e) = 2
To find: equation of hyperbola
Let M be the point on directrix and P(x, y) be any point of hyperbola