**Practice Test – II **

31. If the lines 2x – 3y – 5 = 0 and 3x – 4y = 7 are diameters of a circle of area 154 square units, then the equation of the circle is

(A) x^{2} + y^{2} + 2x – 2y – 62 = 0

(B) x^{2} + y^{2} + 2x – 2y – 47 = 0

(C) x^{2} + y^{2} – 2x + 2y – 47 = 0

(D) x^{2} + y^{2} – 2x + 2y – 62 = 0

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32. The circles x^{2} + y^{2} + x + y = 0 and x^{2} + y^{2} + x – y = 0 intersect at an angle of

(A) π/6

(B) π/4

(C) π/3

(D) π/2

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33. The centres of a set of circles, each of radius 3, lie on the circle x^{2} + y^{2} = 25. The locus of any point with such circle is

(A) 4 ≤ x^{2} + y^{2} ≤ 64

(B) x^{2} + y^{2} ≤ 25

(C) x^{2} + y^{2} ≥ 25

(D) 3 ≤ x^{2} + y^{2} ≤ 9

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34. The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to the circle x^{2} + y^{2} =1 pass through a fixed point

(A) (2, 4)

(B) (–1/2 , –1/4)

(C) (1/2 , 1/4)

(D) (-2 , -4)

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35. Equation of a circle S(x, y) = 0 , ( S(2 , 3) = 16 ) which touches the line 3x + 4y – 7 = 0 at (1, 1) is given by

(A) x^{2} +y^{2} +x +2y –5 =0

(B) x^{2} + y^{2} + 2x + 2y – 6 = 0

(C) x^{2} +y^{2} +4x –6y =0

(D) none of these

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36. If a circle S(x , y) = 0 touches at the point (2 , 3) of the line x + y = 5 and S(1 , 2) = 0 , then radius of such circle

(A) 2 units

(B) 4 units

(C) 1/2 units

(D) 1/√2 units

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37. The number of common tangents that can be drawn to the circle x^{2} + y^{2} – 4x – 6y – 3 = 0 and x^{2} + y^{2} + 2x + 2y + 1 = 0 is

(A) 1

(B) 2

(C) 3

(D) 4

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38. A circle S of radius ‘a’ is the director circle of another circle S_{1} . S_{1} is the director circle of circle S2 and so on. If the sum of the radii of all these circles is 2, then the value of a is

(A) 2 + √2

(B) 2 – √2

(C) 2 – 1/√2

(D) 2 + 1/√2

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39. If the distance of the chord of contact of a circle from any point on its director circle is α , then the radius of the circle is

(A) 2α

(B) √(2α)

(C) √2 α

(D) 2 √α

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40. Let AB be a chord of the circle x^{2} + y^{2} = r^{2} subtending a right angle at the centre. Then the centroid of the triangle PAB as P moves on the circle is

(A) a parabola

(B) an ellipse

(C) a circle

(D) a pair of straight lines

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41. The locus of centres of circles touching the lines x + 2y = 0 and x – 2y = 0 is

(A) xy = 0

(B) x = 1

(C) y = 1

(D) none of these

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42. The range of values of λ for which the circles x^{2} + y^{2} = 4 and x^{2} + y^{2} – 4λx + 9 = 0 have two common tangents is

(A) λ ∈ [-13/8 , 13/8]

(B) λ > 13/8 Or λ < -13/8

(C) λ∈(1, 13/8)

(D) none of these

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43. If the points A (1, 4) and B are symmetrical about the tangent to the circle x^{2} + y^{2} – x + y = 0 at the origin then co-ordinates of B are

(A) (1, 2)

(B) (√2 , 1)

(C) (4, 1)

(D) none of these

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44. The equation of any tangent to the circle x^{2} + y^{2} – 2x + 4y – 4 = 0 is

(A) y = m (x – 1) + 3√(1+m^{2}) – 2

(B) y = mx + 3√(1+m^{2})

(C) y = mx + 3√(1+m^{2})-2

(D) none of these

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45. Two tangents to the circle x^{2} + y^{2} = 4 at the points A and B meet at P (- 4 , 0). The area of the quadrilateral PAOB, where O is the origin is

(A) 4

(B) 6√2

(C) 4√3

(D) none of these

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46. The line λx + μy = 1 is a normal to the circle 2x^{2} + 2y^{2} – 5x + 6y – 1 = 0 if

(A) 5λ – 6μ = 2

(B) 4 + 5μ = 6λ

(C) 4 + 6μ = 5λ

(D) none of these

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47. The equation of the smallest circle passing through the points of intersection of the line x + y = 1 and the circle x^{2} + y^{2} = 9 is

(A) x^{2} + y^{2} + x + y – 8 = 0

(B) x^{2} + y^{2} – x – y – 8 = 0

(C) x^{2} + y^{2} – x + y – 8 = 0

(D) none of these

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48. The members of a family of circles are given by the equation 2 (x^{2}+ y^{2}) + λx -(1 + λ^{2}) y -10 = 0. The number of circles belonging to the family that are cut orthogonally by the fixed circle x^{2} + y^{2} + 4x + 6y + 3 = 0 is

(A) 2

(B) 1

(C) 0

(D) none of these

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49. The equation of the incircle of the triangle formed by the axes and the line 4x + 3y = 6 is

(A) x^{2} + y^{2} – 6x – 6y + 9 = 0

(B) 4 (x^{2} + y^{2} – x – y) + 1 = 0

(C) 4 (x^{2} + y^{2} + x + y) + 1 = 0

(D) none of these

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50. The equation of circumcircle of an equilateral triangle is x^{2} + y^{2} + 2gx + 2fy + c = 0 and one vertex of the triangle is (1, 1). The equation of incircle of the triangle is

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

(B) 4 (x^{2} + y^{2}) + 8gx + 8fy = (1 – g) (1 + 3g) + (1 – f) (1 + 3f)

(C) 4 (x^{2} + y^{2}) + 8gx + 8fy = g^{2} + f^{2}

(D) none of these

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**Click to See All Answers : **

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