**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

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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

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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

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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

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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)

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6. Tangents are drawn from points on circle x^{2} + y^{2} – 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

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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

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

(A) 6

(B) 8

(C) 10

(D) 4

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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

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

(A) 24 sq unit

(B) 48 sq unit

(C) 96 sq unit

(D) none of these

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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) x^{2} + y^{2} = 25

(B) x^{2} + y^{2} + 2x + 2y – 23 = 0

(C) x^{2} + y^{2} – 2x – 2y – 23 = 0

(D) none of these

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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

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

(B) x^{2}/a^{2} + y^{3}/b^{2} = ex/a

(C) x^{2}/a^{2} + y^{3}/b^{2} = ey/b

(D) x^{2}/a^{2} + y^{3}/b^{2} = ex/a + ey/b

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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) x^{2} + y^{2} = a^{2} + b^{2}

(B) x^{2} + y^{2} = a^{2}

(C) x^{2} + y^{2} = (a + b)^{2}

(D) none of these

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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

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26. The eccentricity of ellipse ax^{2} + by^{2} + 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) $\large \sqrt{\frac{b}{a+b}}$

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27. On the ellipse 4x^{2} + 9y^{2} = 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)

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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

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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) a^{2}x^{2} + b^{2}x^{2} = 4x^{2}y^{2}

(B) a^{2}x^{2} + b^{2}y^{2} = 4x^{2}y^{2}

(C) x^{2} + y^{2} = a^{2}

(D) x^{2} + y^{2} = b^{2}

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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

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**Click to See All Answers : **

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

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

**21. C 22. A 23. B 24. B 25. D 26. A 27. C 28. B 29. A 30. B **