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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

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

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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

$straight x squared space equals straight y squared space plus space xy dy over dx$
$straight x squared space equals straight y squared space plus space 3 xy dy over dx$
$straight x squared space equals straight y squared space plus space 2 xy dy over dx$
$straight x squared space equals straight y squared space plus space 2 xy dy over dx$