#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 27 : Hyperbola

#### Pages 13 - 14

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.

Find the equation of the hyperbola whose focus is (0, 3), directrix is x + y − 1 = 0 and eccentricity = 2 .

Find the equation of the hyperbola whose focus is (1, 1), directrix is 3*x* + 4*y* + 8 = 0 and eccentricity = 2 .

Find the equation of the hyperbola whose focus is (1, 1) directrix is 2*x* + *y* = 1 and eccentricity = \[\sqrt{3}\].

Find the equation of the hyperbola whose focus is (2, −1), directrix is 2*x* + 3*y** *= 1 and eccentricity = 2 .

Find the equation of the hyperbola whose focus is (a, 0), directrix is 2*x* − *y* + *a* = 0 and eccentricity = \[\frac{4}{3}\].

Find the equation of the hyperbola whose focus is (2, 2), directrix is x + y = 9 and eccentricity = 2.

Find the eccentricity, coordinates of the foci, equation of directrice and length of the latus-rectum of the hyperbola .

9*x*^{2} − 16*y*^{2} = 144

Find the eccentricity, coordinates of the foci, equation of directrice and length of the latus-rectum of the hyperbola .

16*x*^{2} − 9*y*^{2} = −144

Find the eccentricity, coordinates of the foci, equation of directrice and length of the latus-rectum of the hyperbola .

4*x*^{2} − 3*y*^{2} = 36

3*x*^{2} − *y*^{2} = 4

2*x*^{2} − 3*y*^{2} = 5.

Find the axes, eccentricity, latus-rectum and the coordinates of the foci of the hyperbola 25*x*^{2} − 36*y*^{2} = 225.

Find the centre, eccentricity, foci and directrice of the hyperbola .

16*x*^{2} − 9*y*^{2} + 32*x* + 36*y* − 164 = 0

Find the centre, eccentricity, foci and directrice of the hyperbola .

x^{2} − y^{2} + 4x = 0

Find the centre, eccentricity, foci and directrice of the hyperbola .

x^{2} − 3y^{2} − 2x = 8.

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the distance between the foci = 16 and eccentricity = \[\sqrt{2}\].

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the conjugate axis is 5 and the distance between foci = 13 .

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in the conjugate axis is 7 and passes through the point (3, −2).

Find the equation of the hyperbola whose foci are (6, 4) and (−4, 4) and eccentricity is 2.

Find the equation of the hyperbola whose vertices are (−8, −1) and (16, −1) and focus is (17, −1).

Find the equation of the hyperbola whose foci are (4, 2) and (8, 2) and eccentricity is 2.

Find the equation of the hyperbola whose vertices are at (0 ± 7) and foci at \[\left( 0, \pm \frac{28}{3} \right)\] .

Find the equation of the hyperbola whose vertices are at (± 6, 0) and one of the directrices is *x* = 4.

Find the equation of the hyperbola whose foci at (± 2, 0) and eccentricity is 3/2.

Find the eccentricity of the hyperbola, the length of whose conjugate axis is \[\frac{3}{4}\] of the length of transverse axis.

Find the equation of the hyperboala whose focus is at (5, 2), vertex at (4, 2) and centre at (3, 2).

Find the equation of the hyperboala whose focus is at (4, 2), centre at (6, 2) and e = 2.

If P is any point on the hyperbola whose axis are equal, prove that SP. S'P = CP^{2}.

Find the equation of the hyperbola satisfying the given condition :

vertices (± 2, 0), foci (± 3, 0)

Find the equation of the hyperbola satisfying the given condition :

vertices (0, ± 5), foci (0, ± 8)

Find the equation of the hyperbola satisfying the given condition :

vertices (0, ± 3), foci (0, ± 5)

Find the equation of the hyperbola satisfying the given condition :

foci (± \[3\sqrt{5}\] 0), the latus-rectum = 8

Find the equation of the hyperbola satisfying the given condition :

foci (0, ± 13), conjugate axis = 24

find the equation of the hyperbola satisfying the given condition:

foci (± \[3\sqrt{5}\] 0), the latus-rectum = 8

(vii) find the equation of the hyperbola satisfying the given condition:

foci (± 4, 0), the latus-rectum = 12

find the equation of the hyperbola satisfying the given condition:

vertices (± 7, 0), \[e = \frac{4}{3}\]

find the equation of the hyperbola satisfying the given condition:

foci (0, ± \[\sqrt{10}\], passing through (2, 3).

find the equation of the hyperbola satisfying the given condition:

foci (0, ± 12), latus-rectum = 36

If the distance between the foci of a hyperbola is 16 and its ecentricity is \[\sqrt{2}\],then obtain its equation.

Show that the set of all points such that the difference of their distances from (4, 0) and (− 4,0) is always equal to 2 represents a hyperbola.

#### Page 18

Write the eccentricity of the hyperbola 9*x*^{2} − 16*y*^{2} = 144.

Write the eccentricity of the hyperbola whose latus-rectum is half of its transverse axis.

Write the coordinates of the foci of the hyperbola 9*x*^{2} − 16*y*^{2} = 144.

Write the equation of the hyperbola of eccentricity \[\sqrt{2}\], if it is known that the distance between its foci is 16.

If the foci of the ellipse \[\frac{x^2}{16} + \frac{y^2}{b^2} = 1\] and the hyperbola \[\frac{x^2}{144} - \frac{y^2}{81} = \frac{1}{25}\] coincide, write the value of b^{2}.

Write the length of the latus-rectum of the hyperbola 16*x*^{2} − 9*y*^{2} = 144.

If the latus-rectum through one focus of a hyperbola subtends a right angle at the farther vertex, then write the eccentricity of the hyperbola.

Write the distance between the directrices of the hyperbola *x* = 8 sec θ, *y* = 8 tan θ.

Write the equation of the hyperbola whose vertices are (± 3, 0) and foci at (± 5, 0).

If *e*_{1} and *e*_{2} are respectively the eccentricities of the ellipse \[\frac{x^2}{18} + \frac{y^2}{4} = 1\]

and the hyperbola \[\frac{x^2}{9} - \frac{y^2}{4} = 1\] then write the value of 2 *e*_{1}^{2} + *e*_{2}^{2}.

#### Pages 18 - 20

Equation of the hyperbola whose vertices are (± 3, 0) and foci at (± 5, 0), is

16*x*^{2} − 9*y*^{2} = 144

9*x*^{2} − 16*y*^{2} = 144

25*x*^{2} − 9*y*^{2 }= 225

9*x*^{2} − 25*y*^{2} = 81

If *e*_{1} and *e*_{2} are respectively the eccentricities of the ellipse \[\frac{x^2}{18} + \frac{y^2}{4} = 1\] and the hyperbola \[\frac{x^2}{9} - \frac{y^2}{4} = 1\] , then the relation between *e*_{1} and *e*_{2} is

3 *e*_{1}^{2} + *e*_{2}^{2} = 2

*e*_{1}^{2} + 2 *e*_{2}^{2} = 3

2 *e*_{1}^{2} + *e*_{2}^{2} = 3

*e*_{1}^{2} + 3 *e*_{2}^{2} = 2

The distance between the directrices of the hyperbola *x* = 8 sec θ, *y* = 8 tan θ, is

\[8\sqrt{2}\]

\[16\sqrt{2}\]

\[4\sqrt{2}\]

\[6\sqrt{2}\]

The equation of the conic with focus at (1,* **−*1) directrix along *x* − *y* + 1 = 0 and eccentricity \[\sqrt{2}\] is

xy = 1

2xy + 4x − 4y − 1= 0

x^{2} − y^{2} = 1

2xy − 4x + 4y + 1 = 0

The eccentricity of the conic 9*x*^{2} − 16*y*^{2} = 144 is

\[\frac{5}{4}\]

\[\frac{4}{3}\]

\[\frac{4}{5}\]

\[\sqrt{7}\]

A point moves in a plane so that its distances PA and PB from two fixed points A and B in the plane satisfy the relation PA − PB = k (k ≠ 0), then the locus of P is

a hyperbola

a branch of the hyperbola

a parabola

an ellipse

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis, is

\[\frac{1}{\sqrt{2}}\]

\[\sqrt{\frac{2}{3}}\]

\[\sqrt{\frac{3}{2}}\]

none of these.

The eccentricity of the hyperbola x^{2} − 4y^{2} = 1 is

\[\frac{\sqrt{3}}{2}\]

\[\frac{\sqrt{5}}{2}\]

\[\frac{2}{\sqrt{3}}\]

\[\frac{2}{\sqrt{5}}\]

The difference of the focal distances of any point on the hyperbola is equal to

length of the conjugate axis

eccentricity

length of the transverse axis

Latus-rectum

The distance between the foci of a hyperbola is 16 and its eccentricity is \[\sqrt{2}\], then equation of the hyperbola is

x^{2} + y^{2} = 32

x^{2} − y^{2} = 16

x^{2} + y^{2} = 16

x^{2} − y^{2}^{ }= 32

If *e*_{1} is the eccentricity of the conic 9x^{2} + 4y^{2} = 36 and e_{2} is the eccentricity of the conic 9x^{2} − 4y^{2} = 36, then

*e*_{1}^{2} − *e*_{2}^{2} = 2

2 < *e*_{2}^{2} − *e*_{1}^{2} < 3

*e*_{2}^{2} − *e*_{1}^{2} = 2

*e*_{2}^{2} − *e*_{1}^{2} > 3

If the eccentricity of the hyperbola x^{2} − y^{2} sec^{2}α = 5 is \[\sqrt{3}\] times the eccentricity of the ellipse x^{2} sec^{2} α + y^{2} = 25, then α =

\[\frac{\pi}{6}\]

\[\frac{\pi}{4}\]

\[\frac{\pi}{3}\]

\[\frac{\pi}{2}\]

The equation of the hyperbola whose foci are (6, 4) and (−4, 4) and eccentricity 2, is

\[\frac{(x - 1 )^2}{25/4} - \frac{(y - 4 )^2}{75/4} = 1\]

\[\frac{(x + 1 )^2}{25/4} - \frac{(y + 4 )^2}{75/4} = 1\]

\[\frac{(x - 1 )^2}{75/4} - \frac{(y - 4 )^2}{25/4} = 1\]

none of these

The length of the straight line x − 3y = 1 intercepted by the hyperbola x^{2} − 4y^{2} = 1 is

\[\frac{6}{\sqrt{5}}\]

\[3\sqrt{\frac{2}{5}}\]

\[6\sqrt{\frac{2}{5}}\]

none of these

The foci of the hyperbola 2*x*^{2} − 3*y*^{2} = 5 are

\[( \pm 5/\sqrt{6}, 0)\]

(± 5/6, 0)

\[( \pm \sqrt{5}/6, 0)\]

none of these

The eccentricity the hyperbola \[x = \frac{a}{2}\left( t + \frac{1}{t} \right), y = \frac{a}{2}\left( t - \frac{1}{t} \right)\] is

\[\sqrt{2}\]

\[\sqrt{3}\]

\[2\sqrt{3}\]

\[3\sqrt{2}\]

The equation of the hyperbola whose centre is (6, 2) one focus is (4, 2) and of eccentricity 2 is

3 (x − 6)^{2} − (y −2)^{2} = 3

(x − 6)^{2} − 3 (y − 2)^{2} = 1

(x − 6)^{2} − 2 (y −2)^{2} = 1

2 (x − 6)^{2} − (y − 2)^{2} = 1

The locus of the point of intersection of the lines \[\sqrt{3}x - y - 4\sqrt{3}\lambda = 0 \text { and } \sqrt{3}\lambda + \lambda - 4\sqrt{3} = 0\] is a hyperbola of eccentricity

1

2

3

4

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 27 - Hyperbola

RD Sharma solutions for Class 11 Maths chapter 27 (Hyperbola) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 27 Hyperbola are Standard Equation of a Circle, Latus Rectum, Standard Equation of Hyperbola, Eccentricity, Introduction of Hyperbola, Latus Rectum, Standard Equations of an Ellipse, Eccentricity, Special Cases of an Ellipse, Relationship Between Semi-major Axis, Semi-minor Axis and the Distance of the Focus from the Centre of the Ellipse, Introduction of Ellipse, Latus Rectum, Standard Equations of Parabola, Introduction of Parabola, Concept of Circle, Sections of a Cone.

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