Baixe 65878601 - Conic - Sections - Qns e outras Exercícios em PDF para Engenharia Civil, somente na Docsity!
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
1. If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is :
(A)
(B) 3
(C)
(D) 3
2. The vertices of a hyperbola are at (0, 0) and (10, 0) and one of its foot is at (18, 0). The equation of the hyperbola is :
(A)
x 2 y^2 25 144
(B)
( x − ) y −
2 2 = 1
(C)
x 2 y^2 25
(D)
( x − ) ( y ) −
2 2 = 1
3. If the eccentricities of the hyperbolas
x a
y b
2 2
2 − 2 = 1 &
y b
x a
2 2
2 − 2 = 1 be e & e 1 ,
then,
2 1 e +^ e^2 = (A) 1 (B) 2 (C) 3 (D) None of these
4. If the line y = 2x + c be a tangent to
the ellipse,
x 2 y^2 8 4
(A) ± 4 (B) ± 6 (C) ± 1 (D) ± 8
5. The equation of the tangent to the parabola, y = x^2 - x at the point where x = 1, is : (A) y = − x − 1 (B) y = - x + 1 (C) y = x + 1 (D) y = x - 1 6. If the vertex of a parabola be at origin and directrix be, x + 5 = 0, then its latus rectum is : (A) 5 (B) 10 (C) 20 (D) 40 7. The centre of the ellipse, 4x^2 + 9y^2 - 16x - 54y + 61 = 0, is : (A) (1, 3) (B) (2, 3) (C) (3, 2) (D) (3, 1) 8. Latus rectum of ellipse, 4x^2 + 9y^2 - 8x - 36y + 4 = 0, is :
(A) 83 (B) (^43)
(C)
(D)
9. The locus of the point of intersection
of the lines
x a
y b
= m &
x a
y b
m
where m is a paramater, is always : (A) A circle (B) A parabola (C) An ellipse (D) A hyperbola
10. If the chord joining the points, (at 12 , 2 at 1 ) and (at 22 , 2 at 2 ) of the parabola, y^2 = 4ax passes through the focus of the parabola, then : (A) t 1 t 2 = - 1 (B) t 1 t 2 = 1 (C) t 1 + t 2 = - 1 (D) t 1 - t 2 = 1 11. The focus of the parabola, y^2 = 4y - 4x is : (A) (0, 2) (B) (1, 2) (C) (2, 0) (D) (2, 1)
Conic Sections
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
12. The line l x + my + n = 0 will touch the parabola, if : (A) mn = a l^2 (B) l m = an^2 (C) l n = am^2 (D) mn = a l 13. The line, x cos α + y sin α = p will be
a tangent to the conic,
x a
y b
2 2
2
: (A) p^2 = a^2 sin^2 α + b^2 cos^2 α (B) p^2 = a^2 + b^2 (C) p^2 = b^2 sin^2 α + a^2 cos^2 α (D) None of these
14. The equation of the tangent to the parabola, y^2 = 4x + 5 parallel to the line y = 2x + 7, is : (A) 2x − y − 3 = 0 (B) 2x − y + 5 = 0 (C) 2x + y + 3 = 0 (D) None of these 15. The eccentricity of curve, x^2 − y^2 = 1 is :
(A)
(B)
(C) 2 (D) 2
16. The equation, 14x^2 − 4xy + 11y^2 − 44x − 8y + 71 = 0 represents : (A) A circle (B) An ellipse (C) A hyperbola (D) A rectangular hyperbola 17. The maximum number of normal that can be drawn from a point to a parabola is : (A) 0 (B) 1 (C) 2 (D) 3 18. The centre of the ellipse,
( x + y − ) ( x y)
2 2 = 1, is :
(A) (0, 0) (B) (1, 1) (C) (1, 0) (D) (0, 1)
19. The point on the parabola, y^2 = 8x at which the normal is inclined at 60º to the x - axis has the co-ordinates : (A) (6, − 4 3 ) (B) (6, 4 3 )
(C) (− 6, − 4 3 ) (D) (− 6, 4 3 )
20. The equation of the ellipse whose latus rectum is 8 & whose eccentricity
is
, referred to the principal axes
of co-ordinates, is :
(A)
x 2 y^2 18 32
+ = 1 (B)
x 2 y^2 8 9
(C)
x 2 y^2 64 32
+ = 1 (D)
x 2 y^2 16 24
21. The equation of an ellipse whose focus is (- 1, 1), whose directrix is x − y + 3 = 0 and whose eccentricity
is
, is given by :
(A) 7x^2 + 2xy + 7y^2 + 10x − 10y + 7 = 0 (B) 7x^2 − 2xy + 7y^2 − 10x + 10y + 7 = 0 (C) 7x^2 − 2xy + 7y^2 − 10x − 10y − 7 = 0 (D) 7x^2 − 2xy + 7y^2 + 10x + 10y − 7 = 0
22. The equation of the hyperbola whose foci are (6, 4) & (− 4, 4) & eccentricity 2 is given by : (A) 12x^2 − 4y^2 − 24x + 32y − 127 = 0 (B) 12x^2 + 4y^2 + 24x − 32y − 127 = 0 (C) 12x^2 − 4y^2 − 24x − 32y + 127 = 0 (D) 12x^2 − 4y^2 + 24x + 32y + 127 = 0
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
parabola at the point,
a m
a (^2) m
, is :
(A) y = m^2 x - 2 mx - am^3 (B) m^3 y = m^2 x - 2 am^2 - a (C) m^3 y = 2 am^2 - m^2 x + a (D) None of these
33. If a tangent of y^2 = ax made angle of 45º with the x - axis, then its point of contact will be :
(A)
a a 2 4
^
^
(B) −
^
a a 2 4
(C)
a a 4 2
^
^
(D) −
^
a a 4 2
34. The locus of a foot of perpendicular drawn to the tangent of parabola, y^2 = 4ax from focus is : (A) x = 0 (B) y = 0 (C) y^2 = 2a (x + a) (D) x^2 + y^2 (x + a) = 0 35. The area of triangle formed inside the parabola y^2 = 4x and whose ordinates of vertices are 1, 2 and 4, will be :
(A)
(B)
(C)
(D)
36. If a normal drawn to the parabola, y^2 = 4ax at the point (a, 2a) meets parabola again on (at^2 , 2 at), then the value of t will be : (A) 1 (B) 3 (C) - 1 (D) - 3 37. If (2, 0) is the vertex and y - axis the directrix of a parabola, then its focus is : (A) (2, 0) (B) (- 2, 0) (C) (4, 0) (D) (- 4, 0) 38. The length of chord of contact of the tangents drawn from the point (2, 5) to the parabola, y^2 = 8x, is :
(A)
41 (B) 41
(C)
41 (D) 2
39. The equations of the tangents of the ellipse, 9x^2 + 16y^2 = 144 which passes through the point (2, 3) is : (A) y = 3, x + y = 5 (B) y = - 3, x - y = 5 (C) y = 4, x + y = 3 (D) y = - 4, x - y = 3 40. The length of the latus rectum of the parabola, 9x^2 - 6x + 36y + 19 = 0 is (A) 36 (B) 9 (C) 6 (D) 4 41. The locus of the point of intersection of the perpendicular tangents to the parabola, x^2 = 4ay is : (A) Axis of the parabola (B) Directrix of the parabola (C) Focal chord of the parabola (D) Tangent at vertex to the parabola 42. The point of intersection of tangents at the ends of the latus rectum of the parabola, y^2 = 4x is : (A) (1, 0) (B) (- 1, 0) (C) (0, 1) (D) (0, - 1)
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
43. Let P be a variable point on the ellipse
x a
y b
2 2
2
- 2 = 1, with foci F 1 & F 2. If A is
the ares of the triangle PF 1 F 2 , then maximum value of A is : (A) ab (B) abe
(C)
e a b
(D)
a b e
44. From the point (- 1, 2) tangent lines are draen to the parabola, y^2 = 4x, then the equation of chord of contact is : (A) y = x + 1 (B) y = x - 1 (C) y + x = 1 (D) None of these 45. For the above problem, the area of triangle formed by chord of contact and the tangents is given by : (A) 8 (B) (^8 )
(C) 8 2 (D) None of these
46. The radius of the circle having its centre at (0, 3) & passing through the
foci of the ellipse,
x 2 y^2 16 9
(A) 3 (B) 3.
(C) 4 (D) (^12)
47. If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity, is :
(A)
(B)
(C)
(D) 2 2
48. An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm & 4 cm the necessary length of the string and the distance between the pins respectively in cm, are :
(A) 6, 2 5 (B) 6, 5
(C) 4, 2 5 (D) None of these
49. A man running round a race-course notes that the sum of the distances of two flag-posts from him is always 10 metres and the distance between the flag-posts is 8 metres. The area of the path he encloses in sq. metres is (A) 15 π (B) 12 π (C) 18 π (D) 8 π 50. The locus of the middle points of the chords of hyperbola, 3x^2 − 2y^2 + 4x − 6y = 0 parallel to, y = 2x is : (A) 3x − 4y = 4 (B) 3y − 4x + 4 = 0 (C) 4x − 4y = 3 (D) 3x − 4y = 2 51. Consider a circle with its centre lying on the focus of the parabola, y^2 = 2px such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is :
(A)
p p 2
(B)
p p 2
(C)
^
p p 2
, (D)
^
p p 2
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(C)
( x^ −^ ) y
2 2
/
(D)
( x^ −^ ) y
2 2
/
ANSWERS
1. B 2. B 3. A 4. B 5. D 6. C
7. B 8. A 9. D 10. A 11. A 12.C
13. C 14. B 15. D 16. B 17. D 18.B
19. A 20. C 21. A 22. A 23. A 24.C
25. B 26. D 27. A 28. C 29. C 30.B
31. C 32. C 33. C 34. A 35. D 36.D
37. C 38. C 39. A 40. D 41. B 42.B
43. B 44. B 45. C 46. C 47. D 48.D
49. A 50. A 51. AB 52. A 53. B 54.C
55. A 56. A 57. C 58. D 59. D 60.C
61. A
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