Hyperbola Problems

7/16/2019 Hyperbola Problems

http://slidepdf.com/reader/full/hyperbola-problems 1/16

26 Hyperbola

LEVEL–I

1. Find the equation of the hyperbola with vertices at ( 5, 0) and foci at ( 7, 0).

2. Find the equation of the hyperbola satisfying in the given conditions. Foci (0, )10 , passing through

(2, 3).

3. Find the centre, foci, directrices, length of the latus rectum, length & equations of the axes and the

asymptotes of the hyperbola 10x2 – 9y2 + 32x + 36y – 164 = 0.

4. The hyperbola x2/a2 – y2/b2 = 1 passes through the point of intersection of the lines,

7x + 13y – 87 = 0 & 5x – 8y + 7 = 0 & the latus rectum is 5/232 . Find ‘a’ & ‘b’.

5. In a rectangular hyperbola x2 – y2 = a2, prove that SP. 2CPPS , where S and S are foci, C is the

centre and P is any point on the hyperbola.

6. The centre of a variable rectangular hyperbola lie on a line x + y = 3. A variable circle intersects a

hyperbola in such a way that the mean value of points of intersection is always (3, 5). Find the locus

of the centre of the variable circle.

7. If the normals at four points P (xi, y

i), i = 1, 2, 3, 4 on the rectangular hyperbola xy = c2, meet at the

point Q(h, k), prove that

(i) x1

+ x2

+ x3

+ x4= h

(ii) y1

+ y2

+ y3

+ y4

= k

8. The tangent at P on the hyperbola 1 b

y

a

x2

2

2

2

meets one of the asymptote in Q. Show that the locus

of the mid point of PQ is a similar hyperbola.

9. A variable chord of the hyperbola

2 2

2 2

x y1

a b is tangent to the circle x2 + y2 = c2. Prove that locus of

its mid point is

22 2 2 2

2

2 2 4 4

x y x yc

a b a b

.

10. If the tangent at the point (h, k) to the hyperbola (x2/a2) – (y2/b2) = 1, cuts the auxiliary circle

x2 + y2 = a2 at points whose ordinates y1and y

2show that

1y

1+

2y

1=

k

2.



Hyperbola 27

LEVEL–II

1. If two points P & Q on the hyperbola x2/a2 – y2/b2 = 1 whose centre is C be such that CP is

perpendicular to CQ & a < b, then prove that2222 b

1

a

1

CQ

1

CP

1 .

2. A rectangular hyperbola whose centre is C is cut by any circle of radius r at the four points P, Q, R, S.Prove that CP2 + CQ2 + CR 2 + CS2 = 4r 2.

3. A tangent to the parabola x2 = 4 ay meets the hyperbola xy = k 2 in two points P and Q. Prove that

the middle point of PQ lies on a parabola.

4. The perpendicular from the centre upon the normal on any point of the hyperbolax

a

y

b

2

2

2

21

meets at R. Find the locus of R.

5. Tangents are drawn from the point ,b g to the hyperbola 3x2

- 2y2

= 6 and are inclined at angles

and to the x - axis. If tan .tan 2 , prove that 2 22 7 .

6. Tangents are drawn from any point on the rectangular hyperbola x2 – y2 = a2 – b2 to the ellipse

x2/a2 + y2/b2 = 1. Prove that these tangents are equally inclined to the asymptotes of the hyperbola.

7. If a chord joining the points P a asec , tan b g and Q a asec , tan b g on the hyperbola

x2 - y2 = a2 is a normal to it at P, show that tan tan ( sec ) 4 12 .

8. Show that the locus of the middle points of normal chords of the rectangular hyperbola x2 – y2= a2 is

(y2 – x2)3 = 4a2 x2y2.

9. Chords of the hyperbola x2/a2 – y2/b2 =1 are tangents to the circle drawn on the line joining the foci as

diameter. Find the locus of the point of intersection of tangents at the extremities of the chords.

10. A parallelogram is constructed with its sides parallel to the asymptotes of the hyperbola

x2/a2 – y2/b2 = 1, and one of its diagonals is a chord of the hyperbola; show that the other diagonal

passes through the centre.



28 Hyperbola

IIT JEE PROBLEMS (OBJECTIVE)

A. Fill in the blanks

1. An ellipse has eccentricity 1/2 and one focus at the point P(1/2, 1). Its one directrix is the common

tangent, nearer to the point P, to the circle x2 + y2 = 1 and the hyperbola x2 - y2 = 1. The equationof the ellipse in the standard form is................ [IIT - 96]

B. Multiple choice questions with one or more than correct answer.

1. If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2),R(x

3, y

3), S(x

4, y

4), then

(A) x1

+ x2

+ x3

+ x4

= 0 (B) y1

+ y2

+ y3

+ y4

= 0

(C) x1

x2

x3

x4

= c4 (D) y1

y2

y3

y4

= c4 [IIT - 98]

C. Multiple choice questions with one correct answer.

1. Identify the types of curves with represented by the equation 1r 1

y

r 1

x 22

, where r > 1 is

(A) an ellipse (B) a hyperbola (C) a circle (D) none of these [IIT - 81]

2. The curve described parametrically by, x = t2 + t + 1, y = t2 - t + 1 represents : [IIT - 99]

(A) a pair of straight lines (B) an ellipse(C) a parabola (D) a hyperbola

3. Let P a b( sec , tan ) and Q a b( sec , tan ) , where

2

, be two points on the

hyperbolax

a

y

b

2

2

2

21 . If (h, k) is the point of intersection of the normals at P and Q, then

k is equal to [IIT - 99]

(A)a b

a

2 2(B)

F H G

I K J

a b

a

2 2

(C)a b

b

2 2(D)

F H G

I K J

a b

b

2 2

4. If x = 9 is the chord of contact of the hyperbola x 2 - y2 = 9, then the equation of the corre

sponding pair of tangents, is : [IIT - 99]

(A) 9x2 - 8y2 + 18x - 9 = 0 (B) 9x2 - 8y2 - 18x + 9 = 0(C) 9x2 - 8y2 - 18x - 9 = 0 (D) 9x2 - 8y2 + 18x + 9 = 0

5. For hyperbola 1sin

y

cos

x2

2

2

2

, which of the following remains constant with

change in ‘ ’ [IIT - 2003]

(A) abscissa of vertices (B) abscissa of foci

(C) eccentricity (D) directrix

6. The line 2y6x2 touches the hyperbola 4y2x 22 at [IIT - 2004]

(A) (4,- 6 ) (B) (2,-2 6 ) (C) (-4, 6 ) (D) (-2,2 6 )

7. If e1is the eccentricity of the ellipse 1

25

y

16

x 22

and e2is the eccentricity of the hyperbola passing

through the foci of the ellipse and e1e

2= 1, then equation of the hyperbola is [IIT - 2006]

(A) 116

y

9

x 22

(B) 19

y

16

x 22

(C) 125

y

9

x 22

(D) none of these



Hyperbola 29

IIT JEE PROBLEMS (SUBJECTIVE)

1. For any real t,2

eey,

2

eex

tttt

is a point on the hyperbola x2 – y2 = 1. Show that the

area bounded by this hyperbola and the lines joining its centre to the points corresponding to

t and – t1is t

1. [IIT - 82]

2. A series of hyperbolas is drawn having a common transverse axis of length 2a. Prove that the locusof a point P on each hyperbola, such that its distance from the transverse axis is equal to its

distance from an asymptote, is the curve (x2 - y2)2 = 4x2(x2 - a2). [REE-85]

3. Let ‘p’ be the perpendicular distance from the centre C of the hyperbolax

a

y

b

2

2

2

21 to the

tangent drawn at a point R on the hyperbola. If S and S are the two foci of the hyperbola, then

show that ( )RS RS ab

p

F H G

I K J

2 22

24 1 [REE-89]

4. Two straight lines pass through the fixed points )0,a( and have slopes whose products is p > 0.Show that the locus of the points of intersection of the lines is a hyperbola. [REE-93]

5. Find the equations of the two tangents to the hyperbola xy = 27 which are perpendicular to the

straight line 4x – 3y = 7. Also find the points of contact of these tangents. [REE-93]

6. Determine the constant c such that the straight line joining the points (0, 3) and (5, –2)

is tangent to the curve y = c/(x – 1). [REE-94]

7. Determine the loci of the point which divides a chord with slope 4 of xy = 1 in the ratio 1 : 2.

[IIT - 97]

8. The angle between a pair of tangents drawn from a point P to the parabola y2 = 4ax is 45°. Show that

the locus of the point P is a hyperbola. [IIT - 98]

9. Tangents are drawn from any point on the hyperbola 14

y

9

x 22

to the circle x2 + y2 = 9. Find the

locus of midpoint of the chord of contact. [IIT - 2005]



30 Hyperbola

SET–I

1. Equation of the hyperbola with eccentricity 3/2 and foci at (±2, 0) is

(A)x y2 2

4 9

4

9 (B)

x y2 2

9 4

4

9 (C)

x y2 2

4 91 (D) none of these

2. The foci of the hyperbola 4x2 - 9y2 - 36 = 0 are

(A) ( ) ,11 0 (B) ( ) ,12 0 (C) ( ) ,13 0 (D) 0 12, ( )

3. For the hyperbola whose transverse and conjugate axes are respectively 6 and 4 and whose

centre is at the origin

(A) foci are 13 0,d i (B) latus rectum = 2 7

(C) e = 13 4/ (D) none of these

4. The differential equation of all central conics whose axes are the axes of coordinates is

(A) 1 +dy

dx

2

=d y

dx

2

2 (B) xdy

dxy

d y

dx

2 2

2= y

dy

dx

(C)dy

dxy

d y

dx

2 2

2y = x

dy

dx(D) none of these

5. If the eccentricity of the hyperbola x2 y2 sec2 = 5 is 3 times the eccentricity of the

ellipse x2 sec2 + y2 = 25, then a value of is

(A) /6 (B) /4 (C) /3 (D) /2

6. The line x y pcos sin touches the hyperbolax

a

y

b

2

2

2

21 if

(A) a b p2 2 2 2 2cos sin (B) a b p2 2 2 2cos sin

(B) a b p2 2 2 2 2cos sin (D) a b p2 2 2 2cos sin

7. Tangents drawn from a point on the circle x2 + y2 = 9 to the hyperbolax y2 2

25 161 , then tangents

are at angle

(A) /4 (B) /2 (C) /3 (D) 2 /3

8. The equation of the tangent lines to the hyperbola x2 2y2 = 18 which are perpendicular to

the line y = x are :

(A) y = x ± 3 (B) y = x ± 3

(C)) 2x + 3y + 4 = 0 (D) none of these



Hyperbola 31

9. For all real values of m, the straight line y = mx + 9 42m is a tangent to the curve

(A) 9x2 + 4y2 = 36 (B) 4x2 + 9y2 = 36

(C) 9x2 4y2 = 36 (D) 4x2 9y2 = 36

10. Identify the in correct statement(s) given below in respect of a hyperbola .

(A) the asymptotes to this hyperbolax

a

2

2

y

b

2

2

= 1 are the tangents from its centre .

(B) if the eccentricity of the hyperbola is 5/4 then the eccentricity of its conjugate

hyperbola will be 4/3

(C) no pair of perpendicular tangents can be drawn to hyperbolax2

4

y2

16= 1 from

its point .

(D) the AM of the slopes of the tangents to the hyperbolax2

25

y2

16= 1 through the

point (6 , 2) is 12/11 .

11. A point moves such that the sum of the squares of its distances from the two sides of length'a' of a rectangle is twice the sum of the squares of its distances from the other two sides of

length 'b' . The locus of the point can be

(A) a circle (B) an ellipse

(C) a hyperbola (D) none of these

12. Equation of the hyperbola passing through the point (1, –1) and having asymptotes x + 2y + 3 = 0 and

3x + 4y + 5 = 0 is

(A) 3x2 + 10xy + 8y2 + 14x + 22y + 7 = 0 (B) 3x2 – 10xy + 8y2 + 14x + 22y + 7 = 0

(C) 3x2 – 10xy + 8y2 – 14x + 22y + 7 = 0 (D) none of these

13. An ellipse has eccentricity 1/2 and one focus at the point P (1/2, 1) . Its one directrix is thecommon tangent , nearer to the point P , to the circle x2 + y2 = 1 and the hyperbola x2 y2 = 1.

The equation of the ellipse in the standard form is

(A) x y

13

2

19

2

112

11 (B)

2 2

13

1 19 12

x y 11

(C)

2 213

1 19 12

x y 11

(D) none of these

14. The ellipsex

a

y

b

2

2

2

21 and the hyperbola

x

A

y

B

2

2

2

21 are given to the confocal and length of

minor axis of ellipse is same as the conjugate axis of the hyperbola. If eeand e

hrepresents the eccentricity

of ellipse and hyperbola respectively, then the value of 1 12 2e ee h

is equal to

(A) 1 (B) 2 (C) 4 (D) 6



32 Hyperbola

15. Identify the incorrect statement(s) .

(A) the equation, (x 3)2 + (y + 2)2 =3

2

3 4 1

5

2x y

does not represent a hyperbola

(B) asymptotes to the hyperbolax

a

2

2 y

b

2

2 = 1 are the tangents from the centre .

(C) difference of the focal distances of the point P(3 , 25/4) on the hyperbola,

x

2

16 y

2

25+ 1 = 0 is 8

(D) none of these

16. A variable chord PQ x y p, cos sin of the hyperbolax

a

y

a

2

2

2

221 , subtends a right angle at

the origin. This chord will always touch a circle whose radius is

(A) a (B)a

2(C) a 2 (D) 2 2a

17. Which of the following equations in parametric form can represent a hyperbola, where 't' is

a parameter .

(A) x =a

2t

t

1

& y = b

2t

t

1

(B)tx

a

y

b+ t = 0 &

x

a+

ty

b 1 = 0

(C) x = et – et & y = et et (D) none of these

18. The differential equationdx

dy=

3

2

y

xrepresents a family of hyperbolas (except when it

represents a pair of lines) with eccentricity

(A)3

5(B)

5

3

(C)2

5(D) none of these

19. Two parabolas y2 = 4a(x 1) & x2 = 4a(y

2) always touch each other,

1&

2being

variable parameters . Then their point of contact lies on a

(A) straight line (B) circle (C) parabola (D) hyperbola

20. Two conics a1x2 + 2 h

1xy + b

1y2 = c

1, a

2x2 + 2 h

2xy + b

2y2 = c

2intersect in 4 concyclic

points . Then

(A) (a1 b

1) h

2= (a

2 b

2) h

1(B) (a

1 b

1) h

1= (a

2 b

2) h

2

(C) (a1+ b

1) h

2= (a

2+ b

2) h

1(D) (a

1+ b

1) h

1= (a

2+ b

2) h

2



Hyperbola 33

SET–II

1. For the hyperbolax

a

y

b

2

2

2

2 = 1 the incorrect statement is

(A) the acute angle between its asymptotes is 60º

(B) its eccentricity is 4/3

(C) length of the latus rectum is 2

(D) none of these

2. The equation of common tangent to the curves y2 = 8x and xy = –1 is

(A) 3y = 9x + 2 (B) y = 2x + 1

(C) 2y = x+8 (D) y = x+2

3. The asymptotes of the hyperbola xy = hx + ky are

(A) x k = 0 & y h = 0 (B) x + h = 0 & y + k = 0

(C) x k = 0 & y + h = 0 (D) x + k = 0 & y h = 0

4. The locus of the centre of a circle which touches two given circles externally is(A) ellipse (B) parabola

(C) hyperbola (D) None of these

5. Area of the triangle formed by any tangent of the hyperbola xy = c2, and the coordinate axes, is equal

to

(A) 2c2 (B) 22c

(C) 2 2 c (D) 4c2

6. Total number of tangents of the hyperbola2 2x y

19 4

, that are perpendicular to the line

5x + 2y – 3 = 0, is/are

(A) zero (B) 2

(C) 4 (D) none of these

7. Equations of a common tangent to the two hyperbolasx

a

y

b

2

2

2

2 = 1 and

y

a

x

b

2

2

2

2 = 1 is

(A) 2 2y x a b (B) 2 2y x a b

(C) 2 2

y x a b (D) none of these

8. If the sum of the slopes of the normal from point P to the hyperbola xy = c2 is equal to ( R ) ,

then locus of point ‘P’ is

(A) 2 2x c (B) 2 2

y c

(C) xy = 2c (D) none of these



34 Hyperbola

9. Tangents at any point on the hyperbola2 2

2 2

x y1

a b cut the axes at A and B respectively. If the

rectangle OAPB (where O is origin) is completed then locus of point P is given by

(A)

2 2

2 2

a b1

x y (B)

2 2

2 2

a b1

x y

(C)

2 2

2 2

a b1

y x (D) none of these

10. Number of common tangent to the curves xy = c2 & y2 = 4ax is

(A) 0 (B) 1

(C) 2 (D) 4

11. Shortest distance between the curves

2 2

2 2

x y1

a b , 4x2 + 4y2 = a2 (b > a), is

(A) b

2(B)

b

2

(C)a

2(D)

a

2

12. A conic passes through the point (2, 4) and is such that the segment of any of its tangents at

any point contained between the coordinate axes is bisected at the point of tangency. Then

the foci of the conic are

(A) 2 2 0, & 2 2 0, (B) 2 2 2 2, & 2 2 2 2,

(C) (4, 4) & ( 4, 4) (D) 4 2 4 2, & 4 2 4 2,

13. Locus of the point of intersection of tangents drawn to the curve

2 2

2 2

x y1

a b at the points, whose

sum of eccentric angles is2

, is

(A) x = ab (B) y = ab(C) y = b (D) x = b

14. The co ordinates of a point on the hyperbola,x2

24

y2

18= 1, which is nearest to the line 3x

+ 2y + 1 = 0 is

(A) (3, 6) (B) (6 , 3)

(C) (3, 3) (D) none of these



Hyperbola 35

15. The locus of the mid points of the chords passing through a fixed point ( , ) of the hyperbola,

x

a

2

2 y

b

2

2 = 1 is

(A) a circle with centre 2 2

,

(B) an ellipse with centre 2 2

,

(C) a hyperbola with centre 2 2,

(D) straight line passing through 2 2,

16. The asymptote of the hyperbolax

a

y

b

2

2

2

2 = 1 form with any tangent to the hyperbola a

triangle whose area is a2 tan in magnitude then its eccentricity is

(A) sec (B) cosec (C) sec2 (D) cosec2

17. The tangent to the hyperbola, x2 3y2 = 3 at the point 3 0, when associated with two

asymptotes constitutes(A) isosceles triangle (B) an equilateral triangle

(C) a triangles whose area is 3 sq. units (D) none of these

18. Latus rectum of the conic satisfying the differential equation, x dy + y dx = 0 and passing

through the point (2, 8) is

(A) 4 2 (B) 8

(C) 8 2 (D) 16

19. The portion of the normal between the point P(x, y) of a curve & the x-axis is a constant.Then the curve is

(A) a parabola (B) a circle

(C) a rectangular hyperbola (D) none of these

20. The locus of a point in the Argand plane that moves satisfying the equation,

z 1 + i z 2 i = 3

(A) is a circle with radius 3 & centre at z = 3/2

(B) is an ellipse with its foci at 1 i and 2 + i and major axis = 3

(C) is a hyperbola with its foci at 1 i and 2 + i and its transverse axis = 3

(D) is none of the above



36 Hyperbola

SET–III

Multiple choice question with one or more than one correct answer.

1. The slopes of the common tangent to the hyperbolas16

y

9

x 22

= 1 and16

x

9

y 22

= 1 are

(A) –2 (B) –1 (C) 1 (D) 2

2. The equation of the tangent to the hyperbola 13

y

4

x 22

, parallel to the line y = x + 2, is

(A) y = –x + 1 (B) y = x + 1 (C) y = –x – 1 (D) y = x – 1

3. If the normals at (xi, y

i) i = 1,2 , 3, 4 to the rectangular hyperbola xy = 2 meet at the point (3, 4),

then

(A) x1

+ x2

+ x3

+ x4

= 3 (B) y1

+ y2

+ y3

+ y4

= 4

(C) x1

x2

x3

x4

= –4 (D) y1

y2

y3

y4

= 4

4. If the circle x2

+ y2

= 1 cuts the rectangular hyperbola xy = 1 in four points (x i, yi), i = 1, 2, 3, 4then

(A) x1

x2

x3

x4

= –1 (B) y1

y2

y3

y4

= 1

(C) x1

+ x2

+ x3

+ x4

= 0 (D) y1

+ y2

+ y3

+ y4

= 0

5. The equation2222 )1y(x)1y(x = K will represent a hyperbola for

(A) K (0, 2) (B) K (0, 1) (C) K (1, ) (D) K (0, )

Read the passage given below and answer the questions :

The hyperbola which hasBB

as its transverse axis, andAA

as its conjugate axis, is said to be

the conjugate hyperbola of the hyperbola whose transverse and conjugate axes are respectively

AA and BB . Thus the hyperbola 1a

x

b

y2

2

2

2

is conjugate to the hyperbola 1 b

y

a

x2

2

2

2

.

Two diameters are said to be conjugate when each bisects all chords parallel to the others.

If y = mx, y = m1x be conjugate diameters, then 2

2

1a

bmm . Let y = m

1x + c be a set of chords

parallel to y = m1x, then the diameter y = x

ma

b2

2

bisects them all.

6. Through the positive vertex of the hyperbola at tangent is drawn; then it meet the conjugate

hyperbola at

(A) 2 b,a (B) b,a2 (C) (a, b) (D) none of these



Hyperbola 37

7. If e and e be the eccentricities of a hyperbola and its conjugate, then the value of 22 e

1

e

1

is

(A) 2 (B) 1 (C) 0 (D) none of these

8. A straight line is drawn parallel to the conjugate axis of a hyperbola to meet it and the conjugate

hyperbola in the points P and Q, then the tangents at P and Q meet on the curve

(A) 2

2

2

2

2

2

4

4

a

x4

a

x

b

y

b

y

(B) 2

2

2

2

2

2

4

4

a

x2

a

x

b

y

b

y

(C) 2

2

2

2

2

2

4

4

a

x4

a

x

b

y

b

y

(D) none of these

9. Tangents are drawn to a hyperbola from any point on one of the branches of the conjugate

hyperbola, then their chord of contact will touch the

(A) other branch of the conjugate hyperbola (B) at the point of contact

(C) at the focus (D) none of these

10. If the asymptotes are the straight lines x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and which passes

through the points (1, –1), then the conjugate the hyperbola is

(A) 3x2 + 10xy + 9y2 + 14x + 22y + 23 = 0 (B) 3x2 + 10xy + 8y2 + 14x + 22y + 23 = 0

(C) 3x2 – 10xy + 8y2 + 14x + 22y + 23 = 0 (D) none of these


If an incoming light ray passing through one focus S strike convex side of the hyperbola then it

will get reflected towards other focus S .

11. A ray emanating from the point (5, 0) is incident on the hyperbola 9x2 – 16y2 = 144 at the pointP with abscissa 8, then the equation of the reflected ray after first reflection is

(A) 0315y13x33 (B) 0315y13x33

(C) 0315y13x33 (D) none of these

12. In the above question (6) the point P lies in the first quadrant is

(A) 3,8 (B) 33,8

(C) 33,5 (D) none of these



38 Hyperbola

13. If a variable straight line psinycosx , which is a chord of the hyperbola

)a b(1 b

y

a

x2

2

2

2

. Subtend a right angle at the centre of the hyperbola then it always touches

a fixed circle whose radius is

(A)ab

(b 2a)

(B)

)a b(

ab2

22

(C))a b(

ab

22 (D) none of these

14. PQ and QR are two focal chords of an ellipse and the eccentric angles of P, Q, R are 2,2,2

respectively. Then tantan is equal to

(A) cot (B) 2cot

(C) cot2 (D) none of these


Let the equation of the hyperbola be ax2 + 2hxy + by2 + 2gx + 2fy + c = 0. Now the equation

of the asymptotes differs from that of the hyperbola only by a constant, hence the equation of

the asymptotes is 0cfy2gx2 byxhy2ax 22 , where is to be so chosen that this

may represent a pair of straight lines.

Equation of the hyperbola differs from the equation of the asymptotes by the same constant that

the equation of the asymptotes differs from that of the conjugate hyperbola , that is

Hyperbola + Conjugate Hyperbola = 2 ( Pair of Asymptotes)

while finding the asymptotes the value of can also be obtained by using the fact that the

asymptotes pass through the centre.

15. The pair of asymptotes of the hyperbola 6x2 – 7xy – 3y2 – 2x – 8y – 6 = 0 are

(A) 6x2 – 7xy – 3y2 – 2x – 8y – 4 = 0 (B) 6x2 – 7xy – 3y2 – 2x – 8y – 8 = 0

(C) 6x2 – 7xy – 3y2 – 2x – 8y – 12 = 0 (D) 6x2 – 7xy – 3y2 – 2x – 8y – 16 = 0

16. The equation of the hyperbola, conjugate to the hyperbola given in above question is

(A) 6x2 – 7xy – 3y2 – 2x – 8y – 20 = 0 (B) 6x2 – 7xy – 3y2 – 2x – 8y – 2 = 0

(C) 6x2 – 7xy – 3y2 – 2x – 8y – 14 = 0 (D) 6x2 – 7xy – 3y2 – 2x – 8y – 10 = 0

17. If asymptotes of hyperbola are x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and hyperbola passes through

(1, –1), then its equation is(A) (x + 2y + 3) (3x + 4y + 5) – 6 = 0 (B) (x + 2y + 3) (3x + 4y + 5) – 8 = 0

(C) (x + 2y + 3) (3x + 4y + 5) – 12 = 0 (D) (x + 2y + 3) (3x + 4y + 5) – 14 = 0

18. True or False

(i) The line 5x + 12y = 9 touches the hyperbola x 2 – 9y2 = 2 at the point (5, –4/3)

(ii) The foci of the hyperbola (x2/4) + (y2/12) = 1 coincide with the foci of the ellipse (x2/25) +

(y2/9) = 1.



Hyperbola 39

(iii) If 3x2 + 5xy + 2y2 = 0 is the pair of asymptotes of a hyperbola, then the pair of axes is

5

xy

2

yx 22

.

(iv) The two concentric hyperbolas, whose axes meets at angles of 45º cut at right angle.

(v) The equation 16x2 – 3y2 – 32x – 12y – 44 = 0 represents a hyperbola with centre at (1, –2).

19. Fill in the blanks :

(i) The coordinates of the foci of the hyperbola( )x 1

9

2

( )y 2

16

2

= 1 are ______ and

______ .

(ii) The equation to the locus of the feet of the perpendicular from the focus of the

hyperbola 4x2 9y2 = 36 upon any of its tangent has the equation________.

(iii) The eccentricity of the hyperbola with its principal axes along the coordinate

axes and which passes through (3 , 0) and 3 2 2, is ______ .

(iv) If the foci of the ellipsex y

b

2 2

225 = 1 & the hyperbola

x y2 2

144 81 =

1

25coincide then

the value of b2 is ______ .

(v) The foci of the hyperbola y2 x2 = 1 has the coordinates ______ , ______ and

______ , ______ .

20. Match the column

Column I Column II

(a) Eccentricity of rectangular hyperbola x2 – y2 = a2, is (P) 6

(b) If P is any point on the hyperbola 116

)1y(

9

)1x( 22

,

and S1

and S2

are its foci, then |S1P – S

2P| is equal to (Q)

2

9

(c) The eccentricity of the conjugate hyperbola of hyperbola

x2 – 3y2 = 1 is (R) 2

(d) The latus rectum of the hyperbola

9x2 – 16y2 – 18x – 32y – 151 = 0 is (S) 2

(e) A hyperbola passes through the points (3, 2) and (–17, 12)

and has its centre at origin and transverse axis is along x-axis.

The length of its transverse axis is (T)3

2



40 Hyperbola

LEVEL–I ANSWER

1. 24x2 – 25y2 = 600 2. y2 – x2 = 5

3. (–1, 2); (4, 2) & (–6, 2); 5x – 4 = 0 & 5x + 14 = 0;32

3,

6, 8; y – 2 = 0, x + 1 = 0, 4x – 3y + 10 = 0, 4x + 3y – 2 = 0

4.5

a , b 42

6. x + y = 13

LEVEL–II

4. (x2 + y2)2 (a2y2 - b2x2) = x2y2(a2 + b2)2 9.

2 2

4 4 2 2x y 1a b a b

IIT JEE PROBLEMS (OBJECTIVE)

(A) 1.

xy

F H G

I K J

1

3

1

9

1

1

12

1

2

2b g

(B) 1. ABCD

(C)

1. D 2. C 3. B

4. B 5. B 6. B 7. B



Hyperbola 41

IIT JEE PROBLEMS (SUBJECTIVE)

5. 4y+3x = 36 6. c = 4

7. 16x2 + y2 + 10xy = 2 9.

22222

9

yx

4

y

9

x

SET–I

1. B 2. D 3. C 4. B 5. B

6. D 7. A 8. B 9. D 10. B

11. C 12. A 13. A 14. B 15. C

16. B 17. A 18. B 19. D 20. A

SET–II

1. B 2. D 3. A 4. C 5. A

6. A 7. A 8. A 9. A 10. B

11. C 12. C 13. C 14. B 15. B

16. A 17. C 18. C 19. B 20. D

SET–III

1. BC 2. BD 3. ABC 4. BCD 5. AB

6. A 7. B 8. C 9. A 10. A

11. A 12. B 13. A 14. B 15. A

16. B 17. B

18. (i) F (ii) T (iii) T (iv) T (v) T

19.(i) (–4, 2) and (6, 2) (ii) x2 + y2 = 9 (iii)3

13(iv) b2 = 16

(v) 2,0 and 2,0

20. a-R, b-P, c-T, d-Q, e-S

Date post:	31-Oct-2015
Category:	Documents
Upload:	kishangopi123
View:	144 times
Download:	2 times

Hyperbola Problems

Documents