Tiêu đề XI - maths - chapter 11 - PARABOLA (81-107).pdf ✅

NARAYANAGROUP 81 JEE-MAIN SR-MATHS VOL-IV PARABOLA FOCUS, VERTEX, DIRECTRIX, LATUSRECTUM, AXIS, FOCAL CHORD, FOCAL DISTANCE, DOUBLE ORDINATE 1. The ends of latusrectum of a parabola are (-3,1) and (1,1)then one of the equation of parabola is 1.   2 x y   1 4 2.   2 x y   1 4 3.   2 x y   1 2 4.   2 x y   1 2 2. The length of the latusrectum of the parabola       2 2 2 169 1 3 5 12 17   x y x y         is 1) 14/13 2) 7/13 3) 28/13 4) 56/13 3. The length of double ordinate of parabola 2 y x  8 which subtends an angle 0 60 at vertex is 1. 4 3 2. 8 3 3. 16 3 4.32 3 4. Tangents are drawn from P3,0 to the circle 2 2 x y  1 touches the circle at points A and B. The equation of locus of the point whose distances from the point P and the line AB are equal is 1) 2 9 48 80 0 y x    2) 2 9 48 80 0 y x    3) 2 3 12 39 0 y x    4) 2 3 12 39 0 x y    5. The focus of parabolic mirror is at a distance of 5cm from its vertex if the mirror is 45cm deep, the distance AB 1) 30cm 2) 40cm 3) 50cm 4) 60cm 6. The length of the double ordinate of the parabola 2 y x y     8 6 1 0 which is at a distance of 32 units from vertex is 1) 28 2) 30 3) 32 4) 26 7. If the equation        2 2 2 25 5 3 3 4 1 x y x y       represents a parabola then its axis is 1) 4x+3y-10=0 2) 4x+3y-15=0 3) 4x+3y-29=0 4) 4x+3y-17=0 8. The value of p such that the vertex of 2 y x px    2 13 is 4 units above the x-axis is 1) 2 2) 4 3) 5 4) 3 9. The vertex of a parabola is (2, 0) and its directrix is y-axis. The end of latusrectum in the first quadrant is 1) (2, 4) 2) (4, 4) 3) (6, 4) 4)(8, 4) 10. If two circles 2 2 x y x y      6 6 13 0 and 2 2 x y y     8 9 0 intersects at A and B . The focus of the parabola whose directrix is line AB and vertex at (0, 0) is 1) 3 1, 5 5       2) 3 1, 5 5        3) 3 1 , 5 5         4) 3 1 , 5 5        EQUATION OF PARABOLA / CONIC 11. The equation of the parabola with axis 3 4 4 0 x y    the tangent at the vertex 4 3 7 0 x y    and with length of latusrectum 4 is 1)     2 3 4 4 10 4 3 7 x y x y      2)     2 3 4 4 4 4 3 7 x y x y      3)     2 3 4 4 20 4 3 7 x y x y      4)     2 3 4 4 5 4 3 7 x y x y      12. Let y f x    be a parabola, having its axis parallel to y- axis, which is touched by the line y x  at x 1, then 1) 2 0 '(0) 1 f f     2) f f f 0 ' 0 '' 0 1         3) f ' 1 2    4) f f ' 0 ' 1      13. The focus of a parabola is (1, 2) and the point of intersection of the directrix and axis is (2, 3). Then the equation of the parabola is 1)       2 2 2 1 1 2 5 4 x y x y       2)       2 2 2 1 1 2 5 2 x y x y       3)       2 2 2 1 1 2 5 5 x y x y       4)       2 2 2 1 1 2 5 25 x y x y       LEVEL-II (C.W)
82 NARAYANAGROUP PARABOLA JEE-MAIN SR-MATHS VOL-IV 14. P is a point which moves in the xy plane such that the point P is nearer to the centre of a square than any of the sides. The four vertices of the sqaure are   a a , . The region in which P will move is bounded by parts of parabolas of which one has the equation 1) 2 2 y a ax   2 2) 2 2 x a ay   2 3) 2 2 y ax a   2 4) All of these 15. The number of parabolas passing through the three points (1, 3), (6, 13), (-5, -9) is 1) 3 2) 2 3) 0 4) infinite 16. The graph represented by the equation 2 x t  sin , y t  2cos is 1) a portion of a parabola 2) a parabola 3) a part of sine graph 4) a part of hyperbola TANGENT AND ITS APPLICATIONS 17. The locus of the centroid of triangle formed by a tangent to 2 y x  36 with coordinate axes is 1) 2 y x  9 2) 2 y x  3 3) 2 y x  3 4) 2 y x  9 18. If the line 7 6 13 0 x y    touches the parabola 2 y x y     7 8 14 0 then the point of contact is 1) (2, 1) 2) (1, -1) 3) (-1, 1) 4) (1, 1) 19. The straight line x y k   1 touches the parabola y x x   1  if 1) k  1 2) k  0 3) k 1 4) k takes any real value 20. If two tangents drawn from the point  ,  to the parabola 2 y x  4 be such that the slope of one tangent is double of the other then 1) 2 2 9    2) 2 2 9    3) 2 2 9    4) 2    2 21. Equation of the common tangent to 2 y x  4 and 2 x y  32 is 1) x y    2 4 0 2) x y    2 8 0 3) 2 8 0 x y    4) 2 16 0 x y    22. The locus of point of intersection of the two tangents to 2 y ax  4 inclined at an angle 0 45 is 1) 2 2 2 x y ax a     4 2 0 2) 2 2 2 x y ax a     6 0 3) 2 2 2 x y ax a     8 4 0 4) 2 2 2 x y ax a     2 4 0 23. Area of the triangle formed by the pair of tangents drawn from (-1, 4) to 2 y x  16 and the chord of contact of (–1,4) is 1) 8 2 2) 16 3 3) 5 2 4) 16 2 24. Locus of the point of intersection of perpendicular tangents drawn one each to the parabolas     2 2 y x y x     4 1 , 8 2 is 1) x   12 0 2) x   8 0 3) x   4 0 4) x  3 0 25. The locus of point of intersection of two tangents to 2 y ax  4 at t and 2t on the parabola is 1) 2 2 9 y ax  2) 2 4 9 y ax  3) 2 3 4 y ax  4) 2 3 8 y ax  26. Two tangents to parabola 2 y ax  4 have inclinations 1 and 2 with x-axis such that 2 2 1 2 tan tan     k then the locus of the point of intersection is 1) y kx  2) 2 2 y kx ax   2 3) 2 y ax  2 4) 2 y a  2 27. Equation of the common tangent to the circle 2 2 x y ax   4 and 2 y ax  4 is 1) x y a    0 2) x  0 3) x a  4) x y a    0 28. Equation of the two tangents drawn from (1, 4) to the parabola 2 y x  12 are 1) x y x y       3 0,3 1 0 2) x y x y       1 0, 2 4 0 3) x y x y      2 0, 3 4) x y x y       1 0, 2 4 0 29. The angle between the two tangents drawn from origin to the parabola   2 y a x a   4 is 1) 0 90 2) 0 30 3)   1 tan 2  4) 0 45
NARAYANAGROUP 83 JEE-MAIN SR-MATHS VOL-IV PARABOLA 30. Two tangents to 2 y ax  4 make angles 1 2  , with x axis. If 1 2 cos cos    k then the locus of their intersection is 1)   2 2 2 2 x k x a y        2)   2 2 2 2 x k x a y        3)   2 2 2 2 x k x a y      4   4)   2 2 2 2 4x k x a y        31. If a tangent is drawn to the parabola 2 y x  4 through (-2, 1) then the point of contact is 1) (2, 1) 2) (1, 2) 3) (1, -2) 4) 2, 2  32. The equation of a tangent to the parabola 2 y x  8 is y x   2 , The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is 1) (-2, 0) 2) (-1, 1) 3) (0, 2) 4)( 2, 4) 33. The number of points on the curve 1 2 x y e    from which two perpendicular tangents can be drawn to the parabola 2 x y  4 is equal to 1) 2 2) 3 3) 4 4) 5 34. The equation to the Locus of the point of intersection of the perpendicular tangents to the parabola 2 x x y     4 4 8 0 is 1) y   4 0 2) x  3 0 3) x y   2 4) x  3 0 35. The mirror image of the parabola 2 y x  4 in the tangent to the parabola at the point (1,2) is 1)     2 x y    1 4 1 2)     2 x y    1 4 1 3)    2 x y    1 4 1 4)    2 x y    1 4 1 36. A chord of parabola 2 y ax  4 subtends a right angle at the vertex . The tangents at the extremities of chord intersect on 1. x a   0 2. x a   2 0 3. x a   3 0 4. x a   4 0 37. The point on the parabola 2 y x x    7 2 which is closest to the line y x   3 3 is 1. 2,8 2. 2, 8  3.2,8 4.   2, 8 38. If the distance of two points P and Q on the parabola 2 y ax  4 from the focus of a parabola are 4 and 9 respectively then the distance of the point of intersection of tangents at P and Q from the focus is 1. 8 2. 6 3. 5 4. 13 39. Point of contact of the line kx y    4 0 w.r.t the parabola 2 y x x   is 1. 2,2 2. 2, 2  3. 2,6 4. 2, 6  NORMAL AND ITS APPLICATIONS 40. If a normal subtends a right angle at the vertex of a parabola 2 y ax  4 then its length is 1. 2 3a 2. 3 a 3. 6 3 a 4.8 3 a 41. y x   2 8 2 is a normal chord to 2 y x  8 . Then its length is 1) 16 3 2) 4 3 a 3) 12 3 4) 2 3 a 42. A normal chord of the parabola 2 y x  4 makes an angle 0 45 with the axis of the parabola. Then its length is 1) 8 2 2) 10 2 3) 6 3 4) 4 3 43. The point of intersection of normals to the parabola 2 y x  4 at the points whose ordinates are 4 and 6 is 1) (30, -21) 2) (21, -30) 3) (17, -19) 4) (19, -18) 44. The set of points on the axis of the parabola 2 y y x     2 4 5 0 from which all the three normals to the parabola are real is : 1)  x x ,1 ; 3    2)  x x , 1 ; 1     3)  x x ,3 ; 1    4)  x x , 3 ; 3     45. The normal at P8,8 to the parabola 2 y x  8 cuts it again at Q then PQ= 1) 10 2) 10 5 3) 5 10 4) 50 PROBLEMS ON CHORD 46. The length of the chord to the parabola 2 x ay  4 passing through the vertex and having slope Tan is 1. 4 sec cot a Co   2. 4 tan sec a   3. 4 cos cot a   4. 4 sin tan a  
84 NARAYANAGROUP PARABOLA JEE-MAIN SR-MATHS VOL-IV 47. An equilateral triangle is inscribed in the parabola 2 y ax  4 where one vertex is at the vertex of the parabola. The length of side of triangle 1) 8 3 a 2) 4 3 a 3) 3 3 a 4) 2 3 a 48. A line L passing through the focus of the parabola   2 y x   4 1 intersects the parabola in two distinct points. If ‘m’ be the slope of the line L, then : 1) m R  2)    1 1 m 3) m or m    1 1 4) m  0 49. Let PQ be a chord of the parabola 2 y x  4 . A circle drawn with PQ as diameter passes through the vertex ‘V’ of the parabola. If the area of triangle PVQ is 20 then coordinates of P are 1)   16, 8 2)16,8 3) 16, 8  4) 8,16 50. The triangle PQR of area ‘A’ is inscribed in the parabola 2 y ax  4 such that the vertex P lies at the vertex of the parabola and base QR is a focal chord. The modulus of the difference of the ordinates of the points Q and R is 1) 2 A a 2) A a 3) 2A a 4) 4A a 51. AB is a chord of the parabola 2 y ax  4 with vertex A. BC is drawn  as to AB meets the axis at C. The projection of BC on the axis of the parabola is 1) a 2) 2a 3) 4a 4) 8a 52. Consider the parabola 2 y x  4 ; A 4, 4 ; B 9, 6      be two fixed points on the parabola. Let ‘C’ be a moving point on the parabola between A and B such that the area of the triangle ABC is maximum. The coordinate of C is 1) 1 ,1 4       2) 4, 4 3)3,2 3 4)3, 2 3   53. Minimum area of circle which touches the parabola 2 y x  1 and 2 y x  1 is 1) 9 16 sq units  2) 9 32 sq units  3) 9 8 sq units  4) 9 4 sq units  54. A ray of light travels along a line y=4 and strikes the surface of a curves   2 y x y   4 , then equation of the line along which reflected ray travel is 1) x  0 2) x  2 3) x y   4 4) 2 4 x y   55. PSQ is a focal chord of a parabola whose focus is S and Vertex A, PA and QA are produced to meet the directrix in R and T respectively. Then   RST 1) 0 90 2) 0 60 3) 0 45 4) 0 30 56. The tangents to the parabola 2 y ax  4 at P t 1  and Q t 2  intersect at R. Then the area of PQR is 1)   2 2 1 2 2 a t t  2)   2 1 2 2 a t t  3)   2 3 1 2 2 a t t  4)   2 2 a t t 1 2  57. The circle 2 2 x y x R     2 0,   touches the parabola 2 y x  4 externally, then 1)   0 2)   0 3)  1 4)   0 58. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The road way which is horizontal and 100m long is supported by vertical wires attached to the cable, the longest wire being 30m and the shortest being 6m. The length of a supporting wire attached to the road way 18m from its middle 1) 10m 2) 9.11m 3) 9m 4) 11m 59. M is the foot of the perpendicular from a point P on the parabola   2 y x   8 3 to its directrix and S is the focus of the parabola, if SPM is an equilateral triangle, the length of each side of the triangle is 1) 2 2) 3 3) 4 4) 8