Single answer questions
1. If tanθ1 tanθ2 = , then chord joining two points θ1 and θ2 on the ellipse will subtend a right angle at
2. If the locus of the mid point of the chord of the ellipse drawn parallel to y = m1x is y = m2x then m1m2 =
3. If two points are taken on minor axis of an ellipse at the same distance from the centre as the foci, the sum of the squares of the perpendiculars from these points on any tangent to the ellipse if a > b is.
4. The point on the ellipse x2 + 2y2 = 6 whose distance from the line x + y = 7 is least is
A: (2, 1)
5. If chords of contact of tangent from two points (x1, y1), (x2, y2) to the ellipse are at right angle then ratio of the product of abscissa and ordinates is
A: -16 : 1
6. A point on the ellipse at a distance equal to the mean of the lengths of the semi major axis and semi minor axis from the centre is,
7. A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q the angle between the tangent at 'P' and 'Q' of the ellipse x2 + 2y2 = 6 is.
8. The minimum area of triangle formed by the tangent to the ellipse and coordiante axes is.
A: ab sq. units
9. The arc of the quadrilateral formed by the tangents at the end points of latus recta to the ellipse is (in sq. units)
10. If 2x2 + y2 - 24y + 80 = 0 then maximum value of x2 + y2 is
11. The point of intersection of two ellipses be at the extremities of conjugate diameters of
12. Image of the ellipse in the line x + y = 10 is.
13. Tangents drawn to the ellipse from the point P meet the co-ordinate axes at concyclic points. The locus of the point is
A: x2 - y2 = 27
14. If tangents PQ and PR are drawn from a point on the circle x2 + y2 = 25 to ellipse so that the fourth vertex 'S' of parallelogram PQSR lies on the circum circle of triangle PQR, then the eccentricity of the ellipse is
15. If the curve x2 + 3y2 = 9 subtends an obtuse angle at the point (2α, α) then a possible value of α2 is.
16. An ellipse slides between two perpendicular straight lines x = 0 and y = 0 then, locus of its foci is.
A: A Circle
17. From a point P, perpendicular tangents PQ and PR are drawn to ellipse x2 + 4y2 = 4 locus of circum centre of triangle PQR is.
A: x2 + y2 = (x2 + 4y2)
18. S(3, 4) and S'(9, 12) are the foci of an ellipse and the foot of perpendicular from S to a tangent to the ellipse is (1, -4). Then eccentricity of the ellipse is
19. Equation of largest circle with centre (1, 0) that can be inscribed in the ellipse x2 + 4y2 = 16 is
A: 3x2 + 3y2 - 6x - 8 = 0
20. Let P be a point on the ellipse (a > b) in the first quadrant whose foci are S1 and S2. Then least possible value of circum radius of triangle PS1S2 is
21. The Foci of the ellipse 3x2 - 4xy + 3y2 = 5 are
22. Consider two points A and B on the ellipse circle drawn having segments of tangents at Aand B in between tangents at the two ends of major axis of ellipse as diameter. Then the length of common chord of the circles is
23. P is a point on the ellipse If area , then the radius of the inscribed circle of ∆PF1F2 is (where F1, F2 are Foci)
24. Number of rational points on the ellipse is
A: Infinitely many
25. If A and B are Foci of ellipse (x - 2y + 3)2 + (8x + 4y + 4)2 = 20 and P is any point on it, then PA + PB is
26. Two tangents are drawn to an ellipse from a point P(h, k): If the points in which these tangents meet the axes of the ellipse be concyclic then locus of P is
A: Rectangular hyperbola
27. A(-3, 0), B(3, 0) are two verticies of a triangle whose perimeter is 16. The locus of the incentre of the triangle is
A: x2 + 4y2 = 9
28. C is the centre and A, B are two points on the conic 4x2 + 9y2 - 8x - 36y + 4 = 0 such that ACB = Then
29. If P and P' denote the lengths of the perpendiculars from focus and centre of the ellipse with semi major axis of length 'a' on a tangent and 'r' denotes the focal distance then
A: rP = aP'
30. represents an ellipse with major axis as y-axis and f is decreasing function positive for all a then
A: (-1, 5)
MULTIPLE CORRECT ANSWER TYPE
31. The equations of the common tangents to the curves x2 + 4y2 = 8 and y2 = 4x are
A) x + 2y + 4 = 0 B) x - 2y + 4 = 0
C) 2x + y = 4 D) 2x - y + 4 = 0
A: A, B
32. An ellipse passes through the point (4, -1) and it's axes are along the axes of coordinates. If the line x + 4y - 10 = 0 is tangent to it then its equation can be
C) D) none of these
A: B, C
33. If x + y = 2 is a tangent to the ellipse whose foci are (2, 3), (3, 5) is
A) length of minor axis is 6 B) length of minor axis is 3
C) length of major axis is D) eccentricity is
A: B, C, D
34. AB and CD are two equal and parallel chords of the ellipse Tangents to the ellipse at Aand B intersect at P and at C and D at Q then the line PQ
A) passes through the origin B) is bisected at origin
C) cannot passes through the origin D) is not bisected at origin
A: A, B
35. In a triangle ABC, a = 4, b = c = 2. A point P moves with the triangle such that the square of its distance from BC is half the rectangle contained by its distances from the other two sides. If D be the centre of locus of P then
A) locus of P is an ellipse with eccentricity
B) locus of P is a hyperbola with eccentricity
C) area of the quadrilateral ABCD = sq. units
D) area of the quadrilateral ABCD = sq. units
A: A, C
36. If the normal at any point P on the ellipse meet the major & minor axis at G and g such that PG : pg = 1 : 5 and area of ∆CSB = 10 sq. units. Where C is centre of ellipse. S is focus and B is one of the end of minor axes then,
A) Equation of Ellipse is
B) Equation of Ellipse is
C) Equation of director circle of ellipse is x2 + y2 = 60
D) Equation of director circle of ellipse is x2 + y2 = 30
A: A, C
37. Extremities of the latus recta of the ellipse (a > b) having a given major axes '2a' lies on
A) x2 = a(a - y) B) x2 = a(a + y)
C) y2 = a(a + x) D) y2 = a(a - x)
A: A, B
38. For the ellipses
A) The foci of each ellipses always lies within the other ellipse
B) Their auxiliary circles are the same
C) Their director circles are the same
D) The ellipses encloses the same area
A: B, C, D
39. The equation 3x2 + 4y2 - 18x + 16y + 43 = k represents
A) an empty set if k < 0 ,
B) an ellipse if k > 0 ,
C) a point if k = 0,
D) cannot represent a real pair of lines for any value of k.
A: A, B, C, D
40. Consider the ellipse and f(x) is a positive decreasing function then
A) If the major axis is X-axis then
B) If the major axis is Y-axis then
C) If the major axis is Y-axis then
D) If the major axis is X-axis then
A: A, C
Let R be a point from which a tangent is drawn to an ellipse. If S and S' are foci of ellipse and P is a point of contact of tangent then image of S with respect tangent is collinear with P and S', image of S' is collenear with P and S. Thus is easily understood by the fact that RS + RS' has minimum value at P.
Now from a point Q two tangents QA and AB are drawn to ellipse. Let T and T' be image of S and S' in QA and QB respectively.
41. Which of the following is correct?
A) BT' = AT B) ST = S' T'
C) ST' = S' T D) all of these
42. Which one of the following is incorrect?
C) D) none of thes
43. If ∆QT'S is rotated by an angle θ about Q and is found to be ∆QS'T, then θ can be
C) D) none of thes
A sequence of ellipses E1, E2, E3... En is constructed as follows: Ellipse En is drawn so as to touch ellipse En-1 at the extremities of the major axis of En - 1 and have its foci at the entremities of the minor axes of En - 1.
44. If En-1 is Independent of n then the eccentricity of the ellipse En - 2 is
45. If eccentricity of ellipse En is 'en' then locus of is a
A: rectangular hyperbola
46. If eccentricity of En is independent of n then the locus of the midpoint of chords of slope '-1' of En (if axis of En is along y-axis).
A. If the tangent to the ellipse
x2 + 4y2 = 16 at the point is p 0
normal to the circle x2 + y2 - 8x - 4y = 0
then may be
B. The eccentric angles of a point on the
ellipse x2 + 3y2 = 6 at a distance 2 units q
from the centre of the ellipse is/are.
C. The eccentric angle of intersection of the
ellipse x2 + 4y2 =4 and the parabola x2 + 1= y r
D. If the normal at the point to the
ellipse intersects s
it again at the point Q(Ø) then Ø is
A: A->p, r, B -> r, s, C-> p, D -> q;
A. If vertices of a rectangle of maxium area
inscribed in the ellipse are p
extremities of latus rectum. Then eccentricity of ellipse is
B. If extremities of diameter of the
circle x2 + y2 = 16 are foci of a ellipse, then q
eccentricity of the ellipse, if its size is just
sufficient to contain the circle is
C. If normal at point (6, 2) to the
ellipse passes through its nearest focus (5, 2) r
3 having centre at (4, 2) then its eccentricity is
D. If extremities of latus rectum of the
parabola y2 = 24x are foci of ellipse and if s
2 ellipse passes through the vertex of the parabola
then its eccentricity is
A: A -> q, B-> q, C-> s, D ->p ;
INTEGER ANSWER QUESTIONS
49. If L is the length of the perpendicular drawn from the origin to any normal of the ellipse the maximum value of L is ----
50. A water jet form a fountain reaches its maximum height of 4 meters at a distance 0.5 m from the vertical passing through the point O of water outlet. The height of the jet above the horizontal OX at a distance of 0.75 meters from the point O is k meters then k = ---
51. An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with non part of the circle being outside the ellipse. Then the radius of the circle is ---