The ellipse x2+ 4y2= 4 is inscribed in a rectangle aligned with

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 Multiple Choice QuestionsMultiple Choice Questions

11.

An ellipse is drawn by taking a diameter of the circle (x–1)2 + y2 = 1 as its semiminor axis and a diameter of the circle x2 + (y – 2)2 = 4 as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is

  • 4x2+ y2 = 4

  • x2 +4y2 =8

  • 4x2 +y2 =8

  • 4x2 +y2 =8

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12.

For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is

  • there is a regular polygon with r/R = 1/2

  • there is a regular polygon with straight r over straight R space equals space fraction numerator 1 over denominator square root of 2 end fraction

  • there is a regular polygon with r/R = 2/3

  • there is a regular polygon with r/R = 2/3

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13.

hyperbola passes through the point P(√2, √3) and has foci at (± 2, 0). Then the tangent to this hyperbola at P also passes through the point

  • left parenthesis negative square root of 2 comma space minus square root of 3 right parenthesis
  • left parenthesis 3 square root of 2 space comma space 2 square root of 3 right parenthesis
  • left parenthesis 2 square root of 2 space comma 3 space square root of 3 right parenthesis
  • left parenthesis 2 square root of 2 space comma 3 space square root of 3 right parenthesis
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14.

The eccentricity of an ellipse whose centre is at the origin is 1/2. If one of its directives is x= –4, then the equation of the normal to it at (1,3/2) is

  • x + 2y = 4

  • 2y – x = 2

  • 4x – 2y = 1

  • 4x – 2y = 1

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15.

The ellipse x2+ 4y2= 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then the equation of the ellipse is

  • x2+ 16y2= 16 

  • x2+ 12y2= 16 

  • 4x2+ 48y2= 48 

  • 4x2+ 48y2= 48 


B.

x2+ 12y2= 16 



straight x squared over 16 space plus straight y squared over straight b squared space equals space 1
It space passes space through space left parenthesis 2 comma 1 right parenthesis
So space 4 over 16 space plus 1 over straight b squared space equals space 1
1 over straight b squared space equals space 3 over 4
straight b squared space equals space 4 over 3
rightwards double arrow space straight x squared space plus space 12 straight y squared space equals space 16
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16.

A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2. Then the length of the semi−major axis is 

  • 8/3

  • 2/3

  • 5/3

  • 5/3

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17.

A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is at 

  • (0, 2)

  • (1, 0)

  • (0,1)

  • (0,1)

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18.

Consider a family of circles which are passing through the point (-1, 1) and are tangent to x-axis. If (h, K) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interva

  • 0 < k < 1/2

  • k ≥ 1/2

  • – 1/2 ≤ k ≤ 1/2

  • – 1/2 ≤ k ≤ 1/2

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19.

The differential equation of all circles passing through the origin and having their centres on the x-axis is

  • straight x squared space equals straight y squared space plus space xy dy over dx
  • straight x squared space equals straight y squared space plus space 3 xy dy over dx
  • straight x squared space equals straight y squared space plus space 2 xy dy over dx
  • straight x squared space equals straight y squared space plus space 2 xy dy over dx
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20.

The normal to a curve at P(x, y) meets the x-axis at G. If the distance of G from the origin is twice the abscissa of P, then the curve is a

  • ellipse

  • parabola

  • circle

  • circle

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