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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

గమనిక : ప్రతిభ.ఈనాడు.నెట్లో వచ్చే ప్రకటనలు అనేక దేశాల నుండి, వ్యాపారస్తులు లేదా వ్యక్తుల నుండి వివిధ పద్ధతులలో సేకరించబడతాయి. ఆయా ప్రకటనకర్తల ఉత్పత్తులు లేదా సేవల గురించి ఈనాడు యాజమాన్యానికీ, ఉద్యోగస్తులకూ ఎటువంటి అవగాహనా ఉండదు. కొన్ని ప్రకటనలు పాఠకుల అభిరుచిననుసరించి కృత్రిమ మేధస్సు సాంకేతికతతో పంపబడతాయి. ఏ ప్రకటనని అయినా పాఠకులు తగినంత జాగ్రత్త వహించి, ఉత్పత్తులు లేదా సేవల గురించి తగిన విచారణ చేసి, తగిన జాగ్రత్తలు తీసుకొని కొనుగోలు చేయాలి. ఉత్పత్తులు / సేవలపై ఈనాడు యాజమాన్యానికి ఎటువంటి నియంత్రణ ఉండదు. కనుక ఉత్పత్తులు లేదా సేవల నాణ్యత లేదా లోపాల విషయంలో ఈనాడు యాజమాన్యం ఎటువంటి బాధ్యత వహించదు. ఈ విషయంలో ఎటువంటి ఉత్తర ప్రత్యుత్తరాలకీ తావు లేదు. ఫిర్యాదులు తీసుకోబడవు.

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