Jm S40 Hyperbola 54836

JEE Mains Super40 Revision Series HYPERBOLA   Doubtnut Today Ques No.

Question JEE Mains Super40 Revision Series HYPERBOLA For the hyperbola

1

x cos

2

2

y − α

sin

2

2

= 1

, which of the following remains constant

α

when α varies? (1) eccentricity (2) directrix (3) abscissae of vertices (4) abscissae of foci

 Watch Free Video Solution on Doubtnut

JEE Mains Super40 Revision Series HYPERBOLA The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is : (1) 2

4 3

(2)

4

(3)

√3

2 √3

(4) √3


JEE Mains Super40 Revision Series HYPERBOLA A hyperbola es through the point P (√2, √3) and has foci at ( 3

± 2, 0)

.them

the tangent to this hyperbola at P also es through the point (1) (3√2, 2√3) (2) (2, √2, 3√3)

(3) (√3, √2) (4) (

− √2, − √3)


JEE Mains Super40 Revision Series HYPERBOLA 2 2 Consider the hyperbola H : x − y = 1 and a circle S with centre N (x2 , 0) Suppose that H and S touch each other at a point (P (x1 , y1 ) with x1 > 1 and y1 > 0 The common tangent to H and S at P intersects the xaxis at point M. If (l,m) is the centroid of the triangle ΔP M N then the correct expression is 4

dl

(A)

1 = 1 −

dx1

3x

dl

2

for x1

> 1

1

dx1

3x

2

f

or x1 > 1

(D)

=

dm

1

3(√x

2 1

⎟f

f 3

1

or x1 > 1

(C)

− 1) ⎠

1 =

dy

⎞

x!

dx1

1 = 1 +

(B)

dm

or y1 > 0


JEE Mains Super40 Revision Series HYPERBOLA If 2x 5

− y + 1 = 0

is a tangent to the hyperbola

x a

2

2

y

2

−

= 1

then which of the

16

following CANNOT be sides of a right angled triangle? (a)a, 4, 2 (b) a, 4, 1 (c) 2a, 4, 1 (d) 2a, 8, 1


JEE Mains Super40 Revision Series HYPERBOLA 2 2 lf the eccentricity of the hyperbola x − y (sec)α = 5 is √3 times the eccentricity of 6

the ellipse x2 (sec)

2

α + y

2

= 25,

then a value of α is : (a)

π 6

(b)

π 4

(c)

π 3

(d)

π 2


JEE Mains Super40 Revision Series HYPERBOLA let the eccentricity of the hyperbola

x a

x

7

2

+ 4y

2

= 4.

2

2

y − b

2

2

= 1

if the hyperbola es through a focus of the ellipse then: (a) the

equation of the hyperbola is

x

2

y

2

− 3

= 1

(b) a focus of the hyperbola is (2, 0) (c)

2

the eccentricity of the hyperbola is

√

5

(d) the equation of the hyperbola is

3 x

2

− 3y

2

= 3


be reciprocal to that of the ellipse

JEE Mains Super40 Revision Series HYPERBOLA Area of the quadrilateral formed with the foci x

8

a

2

y −

2

(a

b

2

2

2

2

+ b )

x = 1 and a

(d)

1 (a 2

2

2

2

2

y −

+ b )

b

of

the

hyperbola

2

2

=

− 1

(a)

4(a

2

2

+ b )

(b)

2(a

2

2

+ b )

(c)


JEE Mains Super40 Revision Series HYPERBOLA For each positive integer consider the point P with abscissa n on the curve 2 2 y − x = 1. If dn represents the shortest distance from the point P to the line 9

y = x

then

Lim n→ ∞

(n. dn )

as the value equal to (a)

1 2√ 2

(b)

1 2

(c)

1 √2

(d) 0


10

JEE Mains Super40 Revision Series HYPERBOLA Let the major axis ofa standard ellipse equals the transverse axis of a standard hyperbola and their director circles have radius equal to 2R and R respectively. If e, and e, are the eccentricities ofthe ellipse and hyperbola then the correct relation is (a) 2 2 2 2 2 2 2 2 4e − e = 6 (b) e − 4e = 2 (c) 4e − e = 6 (d) 2e − e = 4 1 2 1 2 2 1 1 2


11

JEE Mains Super40 Revision Series HYPERBOLA 2 An ellipse intersects the hyperbola 2x − 2y = 1 orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then (b) the foci of ellipse are ( ± 1, 0) (a) equation of ellipse is x

2

+ 2y

2

= 2

(d) the foci of ellipse are (t2, 0) (c) equation of ellipse is (x2 2y)


12

JEE Mains Super40 Revision Series HYPERBOLA Find the equation of the chord of the hyperbola 25x2 bisected at the point (5, 3).

− 16y

2

= 400

which is


JEE Mains Super40 Revision Series HYPERBOLA If the foci of the ellipse 13

x

2

y +

16

b

2

2

= 1

and the hyperbola

x

2

y

2

− 144

= 81

1 25

coincide, then find the value of b 2


JEE Mains Super40 Revision Series HYPERBOLA The equation of the transvers and conjugate axes of a hyperbola are, respectively, x + 2y − 3 = 0 and 2x − y + 4 = 0 , and their respective lengths are √2 and 2√ 3.

14

The equation of the hyperbola is

2 5

(x − y − 4)

2

(x + 2y − 3)

2

5 2

2(2x − y + 4)

2

3 −

5

(x + 2y − 3)

− 3(x + 2y − 3)

2

2

= 1

−

3

(2x − y + 4)

2

= 1

5

= 1 2(x + 2y − 3)

2

− 3(2x − y + 4)

2

= 1


15

JEE Mains Super40 Revision Series HYPERBOLA A hyperbola having the transverse axis of length 2 sin θ is confocal with the ellipse 2 2 2 2 2 2 3x + 4y = 12 . Then its equation is x cos ec θ − y sec θ = 1 2 2 2 2 2 2 2 2 2 2 2 2 x sec θ − y cos ec θ = 1 x sin θ − y cos θ = 1 x cos θ − y sin θ = 1


JEE Mains Super40 Revision Series HYPERBOLA Let any double ordinate P N P ' of the hyperbola

x

2

y

25

16

2

−

= 1

be produced on

16 . ′

both sides to meet the asymptotes in QandQ' . Then P QP (c) 41 (d) none of these

Q

is equal to 25 (b) 16


17

JEE Mains Super40 Revision Series HYPERBOLA 2 If S = 0 is the equation of the hyperbola x + 4xy + 3y 2 − 4x + 2y + 1 = 0 , then the value of k for which S + K = 0 represents its asymptotes is 20 (b) − 16 (c) − 22 (d) 18


JEE Mains Super40 Revision Series HYPERBOLA If a ray of light incident along the line 3x + (5 − 4√2)y 18

the hyperbola

x

2

y −

16 x√2 − y + 5 = 0

= 15

gets reflected from

2

= 1

, then its reflected ray goes along the line.

9

(b) √2y

− x + 5 = 0 √2y − x − 5 = 0

(d) none of these


19

JEE Mains Super40 Revision Series HYPERBOLA 2 If the sum of the slopes of the normal from a point P to the hyperbola xy = c is equal to λ(λ ∈ R + ) , then the locus of point P is (a)x2 = λc2 (b) y 2 = λc2 (c) xy = λc

2

(d) none of these


JEE Mains Super40 Revision Series HYPERBOLA The number of points on the hyperbola 20

x a

2

y −

2

b

perpendicular tangents can be drawn to the circle x2 (d) 4

2

2

= 3

+ y

2

from which mutually

= a

2

is/are 0 (b) 2 (c) 3


JEE Mains Super40 Revision Series HYPERBOLA 2 2 The eccentricity of the conic represented by x − y − 21

√2

(c) 2 (d)

2

4x + 4y + 16 = 0

is 1 (b)

1


JEE Mains Super40 Revision Series HYPERBOLA The coordinates of a point on the hyperbola 22

x

2

y

24

line 3x

+ 2y + 1 = 0

are (6, 3) (b) (

2

−

= 1

which s nearest to the

18

− 6, − 3) 6, − 3)

(d) (

− 6, 3)


JEE Mains Super40 Revision Series HYPERBOLA 2 If the line 2x + √6y = 2 touches the hyperbola x − 23

is (

− 2, √6)

(b) (

1

− 5, 2√6) (

1 ,

2

) √6


2y

(d) (4,

2

= 4

, then the point of

− √6)

24

JEE Mains Super40 Revision Series HYPERBOLA 2 2 2 For hyperbola x sec α − y cos ec α = 1, which of the following remains constant with change in ' α' abscissa of vertices (b) abscissa of foci eccentricity (d) directrix


JEE Mains Super40 Revision Series HYPERBOLA If a hyperbola es through the foci of the ellipse

x

2

y

25

25

2

+

= 1

. Its transverse

16

and conjugate axes coincide respectively with the major and minor axes of the ellipse and if the product of eccentricities of hyperbola and ellipse is 1 then the equation of a. hyperbola is

x

2

y

2

− 9

= 1

b. the equation of hyperbola is

16

x

2

y

2

− 9

= 1

c. focus

25

of hyperbola is (5, 0) d. focus of hyperbola is (5√3, 0)


26

JEE Mains Super40 Revision Series HYPERBOLA 2 The tangent to the hyperbola xy = c at the point P intersects the xaxis at T and y axis at T'.The normal to the hyperbola at P intersects the xaxis at N and the yaxis at N' . The areas of the triangles PNT and PN'T' are Δ and Δ' respectively, then 1 Δ

+

1 Δ

'

is (A) equal to 1 (B) depends on t (C) depends on c D) equal to 2


JEE Mains Super40 Revision Series HYPERBOLA The asymptote of the hyperbola 27

x a

2

2

y + b

2

= 1

2

form with ans tangen to the

hyperbola triangle whose area is a2 tan λ in magnitude then its eccentricity is: (a) 2 2 sec λ (b) cos ecλ (c) sec λ (d) cos ec λ


JEE Mains Super40 Revision Series HYPERBOLA Let P (a sec θ, b tan θ)

and Q(a sec cϕ, b tan ϕ)

(where θ

π + ϕ =

be two points

2

on the hyperbola

x a

28

P

and Q

2

2

y − b

2

2

= 1

If (h, k) is the point of intersection of the normals at a

then k is equal to (A)

2

+ b

2

(B) − (

a

a a

2

−(

+ b

2

+ b

2

)

(C)

a

a

2

+ b

2

(D)

b

2

)

b


29

JEE Mains Super40 Revision Series HYPERBOLA 2 2 Consider a branch of the hypetrbolar x − 2y − 2√2x − 4√2y − 6 = 0 with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is (A) 1 − √

2 3

(B) √

3 − 1

(C) 1

2

+ √

2 3

(D) √

3 + 1 2


JEE Mains Super40 Revision Series HYPERBOLA 2 If x = 9 is the chord of  of the hyperbola x − y 2 = 9 then the equation of the corresponding pair of tangents is (A) 9x2 − 8y 2 + 18x − 9 = 0 (B) 30

9x

2

− 8y

2

− 18x + 9 = 0

(C)

9x

2

− 8y

2

− 18x − 9 = 0

(D)

9x^2

8y^2+18x+9=0`


31

JEE Mains Super40 Revision Series HYPERBOLA Tangents are drawn to the hyperbola

x

2

9

2x − y = 1.

y

2

−

= 1 4

parallet to the sraight line

The points of of the tangents on the hyperbola are (A)

2 (

1 ,

)

2√ 2

9

(B) (

1

−

,

√2

)

2√ 2

(C) (3√3,

− 2√2)

(D) (

− 3√3, 2√2)

√2


JEE Mains Super40 Revision Series HYPERBOLA Two conics

x a

32

1 0 < a <

2

2

y − b

(c) a2

2

2

< b

= 1 and x

2

−

. y

intersect, if (a)

0 < b ≤

b 2

(d) a2

< b

2

2

a =

1

(b)

2


JEE Mains Super40 Revision Series HYPERBOLA 2 Tangents drawn from a point on the circle x + 33

x

2

y

2

− 25

= 1,

then tangents are at angle (a)

16

π 4

(b)

π 2

y

2

= 9

(c)

π

to the hyperbola

(d) 2

3

π 3


JEE Mains Super40 Revision Series HYPERBOLA The locus a point P (α, β) moving under the condition that the line y 34

tangent to the hyperbola hyperbola (D) a circle

x a

2

2

y − b

= αx + β

is a

2

2

= 1

is (A) a parabola (B) an ellipse (C) a


35

JEE Mains Super40 Revision Series HYPERBOLA If P is a point on the hyperbola

x

2

y

2

− 7

= 1 and N 3

is the foot of perpendicular

from P on the transverse axis. The tangent to the hyperbola at P meets the transverse axis at T . If O is the centre of hyperbola, then OT . ON = (A) 4 (B) 7 (C) 3 (D) 10


JEE Mains Super40 Revision Series HYPERBOLA 2 If there are two points A and B on rectangular hyperbola xy = c such that abscissa of A = ordinate of B, then locusof point of intersection of tangents at 36 A and B

is (a) y 2

− x

2

= 2c

2

(b) y 2

− x

2

=

c

2

(c) y

= x

2

(d) non of these


JEE Mains Super40 Revision Series HYPERBOLA If e and e' are the eccentricities x

37

a x

2

y −

2

2

b + y

2

2

y = 1 and

2

b

= 1

(b) x

2

2

2

+ y

2

x − a = 2

of

2

2

= 1,

(c) x

then the point (

1

1

,

e 2

+ y

2

= 3

(d) x

2

+ y

the )

hyperbola

lies on the circle (a)

e 2

= 4


JEE Mains Super40 Revision Series HYPERBOLA Find the value of m for which y = mx + 6 is a tangent to the hyperbola x

38

2

y

2

− 100

= 1 49


39

JEE Mains Super40 Revision Series HYPERBOLA 2 2 The tangents from (1, 2√2) to the hyperbola 16x − 25y them an angle equal to:


= 400

include between

JEE Mains Super40 Revision Series HYPERBOLA x

the number of points outside the hyperbola 40

2

y

2

−

= 1

9

from where two

16

perpendicular tangents can be drawn to the hyperbola are: (a) 0 (b) 1 (c) 2 (d) non of these


  Doubtnut to Ask Any Math Question By just a click

 Get A Video Solution For Free in Seconds

 Doubtnut Has More Than 1 Lakh Video Solutions

 Free Video Solutions of NCERT, RD Sharma, RS Aggarwal, Cengage (G.Tewani), Resonance DPP, Allen, Bansal, FIITJEE, Akash, Narayana, VidyaMandir

  Doubtnut Today

Jm S40 Hyperbola 54836

Overview 5o1f4z

More details 6z3438

Related Documents c2h70

Jm S40 Hyperbola 54836

S40 5i682d

Volvo S40 1ew6q

Jm Basic 3n1g24

Jm Etacs_20040203 2g5f3p

Teutonia Jm 641a2k

More Documents from "Rajendra Bisoi" ea1l

Jm S40 Hyperbola 54836

Complete Resonance Mathematics 213239

2015.462792.yoga-dipika.pdf 1do5x

Industrial Drives And Applications-10ee74.pdf 1x3d2j

Advance Heat Transfer _2 5a1rn

Name Change Affidavit 624c44