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PARABOLA

1. If (−2, 5) and (3, 7) are the points of intersection of the tangents and normal at a point on a parabola with the axis of the parabola then the total distance of the point is
A: 

 

2. For each parabola y = x2 + px + q, meeting coordinate, axis at 3 distinct points, if circles are drawn through these points, then the family of circles passes through
A: (0, 1)

 

3. For the parabola y2 = 4ax and the circle x2 + y2 + 2bx = 0 to have more than one common tangents.
A: ab > 0

 

4. Minimum distance between the curves y2 = x − 1 and x2 = y - 1 is equal to
A: 

 

5. The parabola y = x2 − 3x + 8 cuts the X-axis at P and Q. Then lengths of the tangents from (0, 0) to the circle passing through P and Q and also the point (3, 2) is given by
A: 

 

6. If two distinct chords are drawn from the point (4, 4) on the parabola y2 = 4x are bisected on the line y = mx then set of values of m is given by
A: 

 

7. The locus of mid point of family of chords λx + y − 5 = 0 (λ parameter) of the Parabola x2 = 20y

A: x2 = 10(y − 5)
 

8. The Parabola y2 = 8x and the circle x2 + y2 + 2gx + 2fy + c = 0 intersect at four points out of four points, if any three points are conormal, then
A: g  R, f  R, c = 0

 

9. A circle is drawn through any point P on the parabola y2 = 4ax and its foot of perpendicular on the directrix such that the centre of the circle lies on tangent at P. The circle always passes through
A: Vertex of the Parabola

 

10. The diameter of the largest circle is inscribed in the parabola y2 = 4ax and passing through its focus is
A: 8a

 

11. Number of distinct normals that can be drawn from the point   to the parabola y2 = 4x are
A: 2

 

12. The tangent and normal at the extremity of a parabola y2 = 4x form a quadrilateral whose arc is
A: 8

 

13. P (2, 4) and Q are point on the parabola y2 = 8x and the chord subtends right angle at the vertex of the parabola, the coordinates of the point of intersection of normal at P and Q is
A: 

 

14. The points on the axis of the parabola 3y2 + 4y − 6x + 8 = 0 from where 3 distinct normals can be drawn is given by
A: 

 

15. An equilateral triangle SAB is inscribed in the parabola y2 = 4ax having its focus at S. If the chord AB lies to the left of S, then the length of the side of this triangle is
A: 4a(2 − )

 

16. Let the line lx + my = 1 cuts the parabola y2 = 4ax in points A, B. Normals at A and B meet at a point C. Normal from C other than those two meet the parabola at a point D, then D =
A: 
17. PQ is a chord of a parabola which meets axis at R and PQ subtends a right angle at vertex A, then  
A: 

 

18. Consider parabolas y2 = 4ax, x2 = 4ay which have a common tangent at the points P and Q respectively, then the length PQ is
A: 

 

MATRIX MATCH

 

19.

Ans: A s, B r, C q, D p

 

20.


Ans: A s, B q, C r, D p

 

21. 4x + 3y - 8 = 0 is the equation of the chord PQ of the parabola y2 = 4x PS and QS meet the parabola again at R and T respectively where S is the focus, then the equation of RT is
Ans: 8x + 3y = 4

 

22. The segments of the focal chord PSQ of a parabola are the roots of the equation 2x2 − 17x + 7 = 0. The distance between tangent at vertex and directrix of the parabola is
Ans: 

 

23. Equation of circle of minimum radius which touches both the parabolas y = x2 + 2x + 4, x = y2 + 2y + 4 is
Ans: 4x2 + 4y2 − 11x − 11y − 13 = 0

 

24. The triangle formed by the tangent to the parabola y2 = 4x at the point whose abscissa lies in the interval [a2, 4a2], the ordinate and the X - axis, has the greatest area equal to
Ans: 16a3

 

25. ∆ ABC is a right angled triangle (right angled at B) inscribed in parabola y2 = 4x. The minimum length of the intercept cut off by the tangent at A and C to the Parabola on Y-axis is
Ans: 4

 

26. The angle between the tangent drawn from (1, 4) to the parabola y2 = 4x is
Ans: 

 

27. P (2, 3); Q (4, 3) is a focal chord of a parabola whose directrix is 3x + 4y + 4 = 0, then length of PQ is
Ans: 10

 

28. If AFB is a focal chord of the parabola y= 4ax with focus at F and AF = 4, FB = 5, then the latus rectum of the parabola is equal to
Ans: 

 

29. The line y − x + 3 = 0 cuts the parabola y2 = x + 2 at A and B. If P is (, 0), then PA.PB =
Ans: 

 

30. A circle with its centre at the focus of the parabola y2 = 4ax (a > 0) and touching its directrix intersects the parabola at points A, B. The length of AB is equal to
Ans: 4a

 

31. The locus of the mid point of the focal radii of a variable point moving on the parabola y2 = 8x is a parbola whose
   A) Latus rectum is half of the original parabola    B) Vertex is (1, 0)
   C) Directrix is Y - axis                                           D) Focus has the coordinates (2, 0)
Ans: A, B, C, D

 

32. The equations of the common tangents to the parabola y = x2 and y = −(x − 2)2 are
       A) y = 4 (x − 1)       B) y = 0      C) y = − 4(x − 1)       D) y = 30x −50
Ans: A, B

 

33. Locus of the centre of a circle touching a given straight line is the parabola y2 = 8x, then
     A) Center of the given circle is (2, 0)       B) Radius of the given circle is 1 unit
     C) Equation of straight line is x + 1 = 0   D) Equation of given straight line is x − 1 = 0
Ans: A, B, C

 

34. Equation of the tangent to the parabola y2 = 4x which makes an angle 'θ' with its axis is
        A) y = x tan θ + cot θ     B) y = x tan θ + sec θ
        C) x = y cot θ − cot2θ    D) x = y cot θ + tan θ
Ans: A, C

 

35. (xr, yr); r = 1, 2, 3, 4 be the points of intersection of the parabola y2 = 4ax and the circle x2 + y2 + 2gx + 2fy + c = 0, then
    A) y1 + y2 + y3 + y4 = 0     B)  
    C) y1 − y2 + y3 − y4 = 0      D) y1 − y2 − y3 + y4 = 0
Ans: A, B

 

36. If the line x − 1 = 0 is the directrix of the parabola y2 − kx + 8 = 0, the values of k is
     A)      B) −8     C) 4     D) 
Ans: B, C

 

37. If the tangents at A and B on the parabola y2 = 4ax intersect at the point C, then
    A) The ordinate of A, C and B are in A.P.     B) The ordinate of A,B and C are in A.P.
    C) The abscissa of A, C and B are in G.P.     D) The abscissa of A, B, C are in G.P.
Ans: A, C

 

38. Tangent is drawn at any point (x1, y1) other than vertex on the parabola y2 = 4ax. If tangents are drawn from any point on its tangents to the circle x2 + y2 = a2 such that all the chords of contact passes through a fixed point (x2, y2), then
    A) x1, a, x2 in G.P.                      B) 
    C)            D) x1 x2 + y1 y2 = a2   
Ans: B, C, D

 

39. A straight line touches the circle x2 + y2 = 2a2 and the parabola y2 = 8ax. The equation of the line is
      A) y = x + 2a     B) y = −x + 2a     C) y = x − 2a    D) y = x − a
Ans: A, B

 

40. P is a point which moves in the XY-plane such that the point P is nearer to the center of a square than any of the sides. The four vertices of the square are (±a, ±a), the region in which P will move is bounded by parts of parabola of which one has the equation
       A) y2 = a2 + 2ax   B) x2 = a2 + 2ay    C) y2 + 2ax = a2     D) x2 + 2ay + a2 = 0
Ans: A, B, C

 

Paragraph Questions
Passage:
Equation of parabola is y = x2 + ax + 1. Its tangent at the point of intersect of Y-axis and parabola, touches the circle x2 + y2 = r2. It is known that no point on the parabola is below X-axis.

 

41. The radius of circle, when attains maximum value will be
Ans: 

 

42. The slope of tangent, when radius of circle is maximum will be
Ans: 0

 

43. The minimum area bounded by the tangent and the coordinate axes will be
Ans: 
Passage: Let S1 be the parabola having equation y2 − 4ax = 0 through the vertex A of S, two chords AP and AQ are drawn which make an angle  with one another.

 

44. If PQ be a focal chord of the curve, S then the area of the triangle APQ is
Ans: 

 

45. The line PQ always touches a curve S2 whose equation is
Ans: (x − 12a)2 + 8y2 = 128 a2

 

46. If  length of latus rectum of S1 be L1 that of S2 be L2, then  =
Ans: 

Passage: Let two parabolas P1 and P2 are P: y2 = 4ax and P2 : x2 = 4by and a straight lines is L : y = mx + c, then answer the following questions.
 

47. Number of common tangents to P1 and P2 is
Ans: 1

 

48. If line L is a tangent to P1 and P2 both, then
Ans: 

 

49. Locus of midpoint of any focal chord of parabola P2 is
Ans: x2 = 2b(y − b)

 

50. The straight line y = mx + c (m > 0) touches the parabola y2 = 8(x + 2), then the minimum value taken by C is
Ans: 4

 

51. If the condition that the parabolas y2 = 4ax and y2 = 4(x - b) have a common normal other than X - axis (a, b, c being positive reals) is , then least integral value of k is
Ans: 3

 

52. Maximum number of common normal sof y2 = 4ax and x2 = 4by may be equal to
Ans: 5

Posted Date : 19-02-2021

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