MCQ : Ellipse

LEVEL – I

1. The line ax + by + c = 0 is a normal to the ellipse $\large \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$  if

(A) $\large \frac{A^2}{b^2} + \frac{B^2}{a^2} = \frac{(A^2 -B^2)^2}{c^2}$

(B) $\large \frac{A^2}{b^2} + \frac{B^2}{a^2} = \frac{(a^2 -b^2)^2}{c^2}$

(C) $\large \frac{A^2}{b^2} – \frac{B^2}{a^2} = \frac{(A^2 -B^2)^2}{c^2}$

(D) none of these

2. The centre of ellipse $\large \frac{(x+y-4)^2}{16} + \frac{(x-y)^2}{9} = 1$  is

(A) (1, 1)

(B) (2, 2)

(C) (3, 3)

(D) none of these

3. If A and B are two fixed point and P is a variable point such that PA + PB = 4 , then locus of P is

(A) ellipse

(B) circle

(C) dependents upon tangents of PA and PB

(D) depends upon length of AB

4. The set of all values of a for which $\large \frac{x^2}{10-a} + \frac{y^2}{a-11} = 1$   represents an ellipse is

(A) (10, 11)

(B) (- ∞, 10] ∪ (10, ∞)

(C) no value of a

(D) none of these

5. The distance of a point θ on the ellipse $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$   from a focus is

(A) a (1 + e cos θ)

(B) a (e + cos θ)

(C) a (1 – cos θ)

(D) a (1 + 2 e cos θ)

6. Tangents are drawn from points on circle x2 + y2 – 4x – 6y – 12 = 0 where the x coordinate is equal to y coordinate, to the ellipse $\large \frac{(x-2)^2}{9} + \frac{(y-5)^2}{16} = 1$  then the angle between tangents is

(A) π/3

(B) π/4

(C) π/6

(D) none of these

7. Eccentricity of ellipse (5x – 40)2 + (5y – 60)2 = (x + 2y + 17)2 is

(A) 1/2

(B) 1/√3

(C) 1/4

(D) 1/√5

8. If S1 and S2 are the foci of ellipse $\large \frac{(x-2)^2}{32} + \frac{(y-7)^2}{18} = 1$  and a point A≡ (6 , 10) , then AS1 + AS2 is

(A) 6

(B) 8

(C) 10

(D) 4

9. How many tangents can be drawn from point (2, 3) to the ellipse $\large \frac{(x-4)^2}{16} + \frac{(y-4)^2}{9} = 1$

(A) 1

(B) 2

(C) 0

(D) none of these

10. Let P be a variable point on the ellipse $\large \frac{x^2}{100} + \frac{y^2}{64} = 1$  with foci F1 and F2. Then the maximum area of triangle PF1F2 is

(A) 24 sq unit

(B) 48 sq unit

(C) 96 sq unit

(D) none of these

1. B  2. B  3. D  4. C  5. A  6. D  7. D  8. B  9. C  10. C

LEVEL – I

11. A tangent is drawn to the ellipse x2/27 + y2 = 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 x2/9 + y2/5 = 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 $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$  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 $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$  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 x2/16 + y2/9 = 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 x2/4 + y2 = 1 if the difference of the eccentric angle of their point of contact is 2π/3 is

(A) x2/4 + y2 = 4

(B) x2/4 + y2 = 1

(C) 4x2 + y2 = 16

(D) none of these

18. The angle between ellipse x2/4 + y2 = 1 and circle x2 + y2 = 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 x2/4 + y2 = 1 the tangents at the extremities of which intersect at right angle is

(A) 16(x2 + y2)2 = 5(x2 + 4y2)

(B) 16(x2 + y2)2 = 5(x2 + 4y2)2

(C) 5(x2 + y2)2 = 16(x2 + 4y2)2

(D) none of these

20. Angle between tangents drawn from any point on the circle x2 + y2 = (a + b)2 to the ellipse
x2/a + y2/b = a + b is

(A) π/4

(B) tan-1(1/2)

(C) tan-1(1/3)

(D) none of these

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

LEVEL – I
21. The locus of point of intersection of perpendicular tangents of ellipse $\large \frac{(x-1)^2}{16} + \frac{(y-1)^2}{9} = 1$   is

(A) x2 + y2 = 25

(B) x2 + y2 + 2x + 2y – 23 = 0

(C) x2 + y2 – 2x – 2y – 23 = 0

(D) none of these

22. If α, β are the eccentric angles of the extremities of a focal chord of an ellipse, then eccentricity of the ellipse is

(A) $\large \frac{sin\alpha + sin\beta}{sin(\alpha + \beta)}$

(B) $\large \frac{cos\alpha + cos\beta}{cos(\alpha + \beta)}$

(C) $\large \frac{sin(\alpha + \beta)}{sin\alpha + sin\beta}$

(D) none of these

23. Locus of mid-point of the focal chord of ellipse $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$   with eccentricity e is

(A) x2 + y2 = (a+b)2/4

(B) x2/a2 + y3/b2 = ex/a

(C) x2/a2 + y3/b2 = ey/b

(D) x2/a2 + y3/b2 = ex/a + ey/b

24. The locus of foot of perpendicular from focus of ellipse $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$   to its tangent is

(A) x2 + y2 = a2 + b2

(B) x2 + y2 = a2

(C) x2 + y2 = (a + b)2

(D) none of these

25. The length of common chord of ellipse $\large \frac{(x-10)^2}{100} + \frac{(y-21)^2}{81} = 1$
and circle (x – 10)2 + (y – 17)2 = 16 is

(A) 9 units

(B) 10 units

(C) 19 units

(D) none of these

26. The eccentricity of ellipse ax2 + by2 + 2fx + 2gy + c = 0, if major axis is parallel to x- axis, is

(A) $\large \sqrt{\frac{b-a}{b}}$

(B) $\large \sqrt{\frac{a-b}{b}}$

(C) $\large \sqrt{\frac{a}{a+b}}$

(D) $\large \sqrt{\frac{b}{a+b}}$

27. On the ellipse 4x2 + 9y2 = 1, the point at which the tangents are parallel to the line 8x-9y=0 is

(A) (2/5 , 1/5)

(B) (-2/5 , -1/5)

(C) (2/5 , -1/5)

(D) (-2/5 , 1/5)

28. In a ellipse distance between the foci is 10 and distance between directrices is 40 then the length of major and minor axes are

(A) 10 and 5√3

(B) 20 and 10√3

(C) 30 and 20

(D) none of these

29. The locus of mid-point of the portion of the tangents to the ellipse $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$      intercepted between the axes is

(A) a2x2 + b2x2 = 4x2y2

(B) a2x2 + b2y2 = 4x2y2

(C) x2 + y2 = a2

(D) x2 + y2 = b2

30. If α, β be the eccentric angles of the ends of a focal chord of the ellipse  $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$    , then tanα/2 tanβ/2 is equal to

(A) (1-e)/(1+e)

(B) (e-1)/(e+1)

(C) (e+1)/(e-1)

(D) none of these