**QUIZ – I **

31. Equation of the latus rectum of the hyperbola (10x – 5)^{2} + (10y – 2)^{2} = 9(3x + 4y –7)^{2} is

(A) y – 1/ 5 = –3/4( x- 1/ 2)

(B) x – 1/ 5 = –3/4(y – 1/ 2)

(C) y + 1/ 5 = –3/4( x + 1/ 2)

(D) x + 1/ 5 = –3/4(y+1/ 2)

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32. If PN is the perpendicular drawn from a point P on xy = c^{2} to its asymptote, then locus of the mid-point of PN is

(A) circle

(B) parabola

(C) ellipse

(D) hyperbola

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33. Asymptotes of the hyperbola xy = 4x + 3y are

(A) x = 3 , y = 4

(B) x = 4 , y = 3

(C) x = 2 , y = 6

(D) x= 6 , y = 2

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34. The distance between foci of a hyperbola is 16 and its eccentricity is √2, then the equation of hyperbola is

(A) x^{2} – y^{2} = 3

(B) x^{2} – y^{2} = 16

(C) x^{2} – y^{2} = 15

(D) x^{2} – y^{2} = 32

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35. The point of intersection of tangents at ‘ t_{1} ‘ and ‘ t_{2} ‘ to the hyperbola xy = c^{2} is

(A) $\large (\frac{c t_1 t_2}{t_1 + t_2} , \frac{c }{t_1 + t_2} )$

(B) $\large (\frac{2c t_1 t_2}{t_1 + t_2} , \frac{2c }{t_1 + t_2} )$

(C) $\large (\frac{ t_1 t_2}{c(t_1 + t_2)} , \frac{t_1 + t_2 }{c} )$

(D) none of these

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36. The equation 2x^{2} + 3y^{2} – 8x – 18y + 35 = 0 represents

(A) a point

(B) parabola

(C) hyperbola

(D) ellipse

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37. A circle cuts two perpendicular lines so that each intercept is of given length. The locus of the centre of the circle is a conic whose eccentricity is

(A) 1

(B) 1/√2

(C) √2

(D) none of these

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38. The locus of the point of intersection of tangents at the extremities of the chords of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ which are tangents to the circle drawn on the line joining the foci as diameter is

(A) $\frac{x^2}{a^2}-\frac{y^2}{b^2} = \frac{1}{a+b} $

(B) $\frac{x^2}{a^4}+\frac{y^2}{b^4} = \frac{1}{a^2+b^2} $

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

(D) x^{2} – y^{2} = a^{2} + b^{2}

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39. If the portion of the asymptote between centre and the tangent at the vertex of hyperbola$\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ in the third quadrant is cut by the line y + λ(x + a) = 0 ; λ being parameter, then

(A) λ ∈ R^{+}

(B) λ ∈ R^{–}

(C) λ ∈ (0, 1)

(D) none of these

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40. The equation of the common tangent to the curves y^{2} = 8x and xy = –1 is

(A) 3y = 9x + 2

(B) y = 2x + 1

(C) 2y = x + 8

(D) y = x + 2

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41. Let S_{1} and S_{2} be the foci of a rectangular hyperbola, which has the centre at Q, then for any point P on the hyperbola S_{1}P . S_{2}P equals to

(A) S_{1} . S_{2}^{2}

(B) QS_{1}^{2}

(C) QP^{2}

(D) 4QP^{2}

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42. The values of ‘m’ for which a line with slope m is common tangent to the hyperbola

$\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ (a ≠ b) and parabola y^{2} = 4ax is

(A) m ∈ (0, ∞)

(B) m ∈ (-∞, -1) ∪ (1, ∞) -{±√(1+√5)/2}

(C) m ∈(-∞, 2) – √(1+√5)/2

(D) none of these

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43. The chord PQ of the hyperbola xy = c^{2} meets the x-axis at A. If R be the mid point of PQ and O be the centre of the hyperbola, then the triangle ARO is necessarily

(A) equilateral

(B) isosceles

(C) right angled

(D) obtuse angled

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44. Minimum length of a normal chord to the hyperbola xy = c^{2} lying between different branches is

(A) √2 c

(B) 2c

(C) 2 √2 c

(D) none of these

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45. If the tangent at the point P(h , k) on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ cuts the circle x^{2} + y^{2} = a^{2} at the points Q(x_{1}, y_{1}) and R(x_{2}, y_{2}), then = 1/y_{1} + 1/y_{2}

(A) 2/k

(B) 1/k

(C) a/k

(D) b/k

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46. The reflection of the curve xy = 1 in the line y = 2x is the curve 12x^{2} + r xy + sy^{2} + t = 0, then the value of r is

(A) – 7

(B) 25

(C) – 175

(D) none of these

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47. The angle between lines joining the origin to the points of intersection of the line √3x + y = 2 and the curve y^{2} – x^{2} = 4 is equal to

(A) tan^{-1}(2/√3)

(B) π/6

(C) tan^{-1}(√3/2)

(D) π/2

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48. The number of real points at which the line 3y – 2x = 14 cuts the hyperbola $\frac{(x-1)^2}{9}-\frac{(y-2)^2}{4} = 5 $ is

(A) 2

(B) 3

(C) 4

(D) none of these

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49. The line 2x + y = 0 passes through the centre of a rectangular hyperbola, one of whose asymptotes is x – y = 1. The equation of the other asymptote is

(A) 3x + 3y + 1 = 0

(B) 3x + 3y – 1 = 0

(C) x – 2y = 0

(D) none of these

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50. A line y = 2x + 5 intersects a hyperbola only at a point (-2 , 1). The equation of one of its asymptotes is 3x + 2y + 1 = 0. If hyperbola passes through a point (-1 , 0), then the equation of hyperbola is

(A) 6x^{2} + xy – 2y^{2} + 35x – 21y + 29 = 0

(B) 6x^{2} + xy – 2y^{2} – 35x – 21y + 29 = 0

(C) 6x^{2} + xy – 2y^{2} + 35x – 21y – 29 = 0

(D) none of these

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**Click to See Answer : **

**31. A 32. D 33. A 34. D 35. B 36. A 37. C 38. B 39. A 40. D**

**41. C 42. B 43. B 44. C 45. A 46. A 47. C 48. D 49. A 50. D **