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 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
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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 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
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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 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
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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
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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
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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 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
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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
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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
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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
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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
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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) x2 + y2 = 25
(B) x2 + y2 + 2x + 2y – 23 = 0
(C) x2 + y2 – 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) 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
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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) x2 + y2 = a2 + b2
(B) x2 + y2 = a2
(C) x2 + y2 = (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 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) $\large \sqrt{\frac{b}{a+b}}$
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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)
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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) a2x2 + b2x2 = 4x2y2
(B) a2x2 + b2y2 = 4x2y2
(C) x2 + y2 = a2
(D) x2 + y2 = b2
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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 :
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