**LEVEL – I **

11. A tangent is drawn to the ellipse x^{2}/27 + y^{2} = 1 at the point (3√3cosθ , sinθ) where 0 < θ < π/2 . The sum of intercepts of the tangent with the coordinate axes is least when θ equals

(A) π/6

(B) π/3

(C) π/8

(D) π/4

12. The area of the quadrilateral formed by the tangents to the ellipse x^{2}/9 + y^{2}/5 = 1 at the ends of the latus rectum is

(A) 27/4

(B) 9

(C) 27/2

(D) 27

13. P is any point on the ellipse and S and S’ are its foci, then maximum value of the angle SPS’ is

(A) π/4

(B) π/2

(C) 2 tan^{-1}(ae/b)

(D) none of these

14. The curve with parametric equations x = 1 + 4 cosθ , y = 2 + 3 sin θ is

(A) an ellipse

(B) a parabola

(C) a hyperbola

(D) a circle

15. The eccentricity of an ellipse whose latus rectum is half of its major axis is

(A) 1/√2

(B) √(2/3)

(C) √3/2

(D) none of these

16. If base of a triangle is the major axis of the ellipse x^{2}/16 + y^{2}/9 = 1 and third vertex moves on the ellipse, then maximum area of triangle will be

(A) 6

(B) 72

(C) 12

(D) none of these

17. The locus of the point of intersection of the tangents drawn to the ellipse x^{2}/4 + y^{2} = 1 if the difference of the eccentric angle of their point of contact is 2π/3 is

(A) x^{2}/4 + y^{2} = 4

(B) x^{2}/4 + y^{2} = 1

(C) 4x^{2} + y^{2} = 16

(D) none of these

18. The angle between ellipse x^{2}/4 + y^{2} = 1 and circle x^{2} + y^{2} = 2 is θ , then tanθ is equal to

(A) 1/2

(B) 1/√2

(C) 1/2√2

(D) none of these

19. The locus of mid points of chords of an ellipse x^{2}/4 + y^{2} = 1 the tangents at the extremities of which intersect at right angle is

(A) 16(x^{2} + y^{2})^{2} = 5(x^{2} + 4y^{2})

(B) 16(x^{2} + y^{2})^{2} = 5(x^{2} + 4y^{2})^{2}

(C) 5(x^{2} + y^{2})^{2} = 16(x^{2} + 4y^{2})^{2}

(D) none of these

20. Angle between tangents drawn from any point on the circle x^{2} + y^{2} = (a + b)^{2} to the ellipse

x^{2}/a + y^{2}/b = a + b is

(A) π/4

(B) tan^{-1}(1/2)

(C) tan^{-1}(1/3)

(D) none of these

__ANSWER:__

**11. A 12. D 13. C 14. A 15. A 16. C 17. A 18. B 19. B 20. D **