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Conic Sections Test - 5

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Conic Sections Test - 5
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  • Question 1
    2 / -0.83

    If the line 2x –y + λ = 0 is a diameter of the circle  x2 +y2 +6x −6y+5 = 0 then  λ=

    Solution

    x2 + y2 + 6x −6y + 5 = 0
    Center O = (-3, 3)
    radius r = √{(-3)2 + (3)2 - 5} 
    = √{9 + 9 - 5} 
    = √13
    Since diameter of the circle passes through the center of the circle.
    So (-3, 3) satisfies the equation 2x –y + λ= 0
    =>-3*2 - 3 + λ= 0
    =>-6 - 3 + λ= 0
    =>-9 + λ= 0
    =>λ= 9

  • Question 2
    2 / -0.83

    Length of common chord of the circles  x2 +y2 +2x+6y = 0 and  x2 +y2 −4x −2y −6 = 0 is  

    Solution

    Given equation of circles are  

    S1  = x2  + y2  + 2x + 6y = 0 ..................1

    S2  = x2  + y2  −4x −2y −6 = 0 ............2

    Subtract equation 1 - equation 2, we get

    S1  - S2  = 6x + 8y + 6 = 0

    => 6x + 8y + 6 = 0 ............3

    Cente of the circle S2  is (2, 1)

    Now, the length of the perpendicular from the center (2, 1) of of the circle 2 upon the common chord 3 is

    l = (6*2 + 8*1 + 6)/√{62  + 82  }

    => l = (12 + 8 + 6)/√{36 + 64}

    => l = 26/√100

    =>l = 26/10

    =>l = 13/5

    Radius of the circle 2 is

    r = √{22  + 12  - (-6)}

    => r = √{4 + 1  + 6}

    => r = √11

    Now length of common chord = 2 √{r2  - l2  }

     = 2 √{(√11)2  - (13/5)2  }

    = 2 √{11 - 169/25}

    = 2 √{(275 - 169)/25}

    = 2 √{106/25} unit

  • Question 3
    2 / -0.83

    The circles  x2 +y2 +6x+6y = 0 and  x2 +y2 −12x −12y = 0

    Solution

    Given equation of circles are
    x2 +y2 +6x+6y=0....(i)
    and x2 +y2 −12x −12y=0....(ii)
    Here, g1 = 3,f2 = 3, g2 = −6 and f2 = −6
    ∴Centres of circles are C1 (−3,−3) and C2 (6,6) respectively and radii are r1 = 3 √2 and r2 = 6 √2 respectively.
    Now, C1 C2 = √[(6+3)2 + (6+3)2 ]
    = 9 √2
    and r1 + r2
    ​= 3(2)1/2 + 6(2)1/2
    = 9(2)1/2
    ​⇒C1 C2 = r1 + r2
    ∴Both circles touch each other externally.

  • Question 4
    2 / -0.83

    The equation  x2 +y2  = 0 represents

  • Question 5
    2 / -0.83

    The equation 3  (x2 +y2 )+5x −7y −2 = 0 represents

  • Question 6
    2 / -0.83

    Circumcentre of the triangle, whose vertices are (0, 0), (6, 0) and (0, 4) is

  • Question 7
    2 / -0.83

    If  (x2 −a)2 +(y −b)2  = c2  represents a circle, then

    Solution

    The standard  equation of a circle  is given by:

    (x-a)2 + (y-b)2 = c2

    Where (a,b) is the coordinates of center of the circle and c is the radius. 
    if c become zero then the radius of circle become zero then there will be no circle exist hence the radius can not be zero. So the correct option is D
     

  • Question 8
    2 / -0.83

    The length of the chord joining the point (4 cos  θ, 4 sin  θ) and 4 (cos(θ+60o ), 4 sin(θ + 60o )) of the circle  x2 +y2  = 16 is

  • Question 9
    2 / -0.83

    Two perpendicular tangents to the circle  x2 +y2  = r2 meet at P. The locus of P is

    Solution

  • Question 10
    2 / -0.83

    The number of tangents to the circle  x2 +y2 −8x −6y+9 = 0, which pass through the point (3, - 2), is

  • Question 11
    2 / -0.83

    The value of k, such that the equation = 2x2 +2y2 −6x+8y+k = 0 represents a point circle, is

  • Question 12
    2 / -0.83

    The equation  ax2 +by2 +2hxy+2gx+2fy+c = 0 represents a circle only if

    Solution

    ax2 +by2 +2hxy+2gx+2fy+c = 0
    In the circle equation,
    Coefficients of x2 &y2 are the same and non-zero.
    Therefore, a = b ≠0
    There is no term for xy. Therefore, h = 0.
    Radius of the circle: r2 = g2 + f2 –c,
    So, g2 + f2 –c >0

  • Question 13
    2 / -0.83

    x2 +y2 −6x+8y −11 = 0 is a circle. The points (0, 0) and (1, 8) lie

  • Question 14
    2 / -0.83

    The length of tangent from the point (2, - 3) to the circle  2x2 +2y2  = 1 is

    Solution

    Length of tangent from an external point (h,k) to  S = √(S(h,k))
    S(2,-3) = √[2(2)2 + 2(-3)2 - 1] 
    =>S(2,-3) = √[8 + 18 - 1]  
    =>S(2,-3) = √25
    =>S(2,-3) = 5

  • Question 15
    2 / -0.83

    Which one of the following lines is farthest from the centre of the circle  x2 +y2  = 10?

  • Question 16
    2 / -0.83

    Which of the following lines is a normal to the circle  (x −1)2 +(y −2)2  = 10

  • Question 17
    2 / -0.83

    A circle with its centre on the line y = x + 1 is drawn to pass through the origin and touch the line y = x + 2. The centre of the circle is

  • Question 18
    2 / -0.83

    Four distinct points  (2 λ,3 λ),(1,0),(0,1) and (0, 0) lie on a circle for

  • Question 19
    2 / -0.83

    The locus of the point of intersection of the lines x cos  α + y sin  α= a  and x sin  α - y cos  α = b is

  • Question 20
    2 / -0.83

    The line y = m x + c is a normal to the circle  x2 +y2 +2gx+2fy+c = 0 if

  • Question 21
    2 / -0.83

    The line 3x –4y = 0

  • Question 22
    2 / -0.83

    The equations x = a cos θ + b sin  θ , and  y = a sin θ−b cos θy = a sin ⁡θ−b cos ⁡θ , 0  ≤ θ≤2  π represent

  • Question 23
    2 / -0.83

    The number of points which have the same power w.r.t. two (different) concentric circles is

  • Question 24
    2 / -0.83

    A circle passes through (0, 0) ( a, 0), (0, b). The coordinates of its centre are

  • Question 25
    2 / -0.83

    The focus of the parabola  x2 −8x+2y+7 = 0 is

    Solution

     Parabola is x2 –8x + 2y + 7 = 0  
    ∴ (x –4)2 = –2y –7 + 16  
    ∴ (x –4)2 = –2[y –(9/2)]
     ∴ x2 = –4ay  
     ⇒x = x –4, y = y –(9/2) and 2 = 4a
     i.e. a = (1/2) 
     Its focus is given by x = 0 and y = 0 i.e. x –4 = 0  and   y –(9/3) = 0  
    ∴  x = 4   and y = (9/2) 
    ∴focus [4, (9/2)].

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