﻿ 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 | Conic Section

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# Conic Section

#### Multiple 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

• 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

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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

C.

4x – 2y = 1

Eccentricity of ellipse =1/2

Now, a/e = -4
⇒ a = 4 x (1/2) = 2
therefore, b2 = a2(1-e2)
= a2 (1-(1/4)) = 3

Equation of normal at (1, 3/2)
y-3/2 = 2(x – 1)
⇒ 2y – 3 = 4x – 4
⇒ 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

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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

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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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