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

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 

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

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

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

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

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 

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

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)

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

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

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