1
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The eccentricity of an ellipse whose centre is at the origin is ½

The eccentricity of an ellipse whose centre is at the origin is ½. If one of its directrices is x = -4, then the equation of the normal to it at (1, 3/2) is

- 2y - x = 2
- 4x - 2y = 1
- 4x + 2y = 7
- x + 2y = 4

2
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Let k be an integer such that the triangle with vertices (k, –3k), (5, k) and (–k, 2)

Let k be an integer such that the triangle with vertices (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point

- 2, -1/2
- 1, 3/4
- 1, -3/4
- 2, 1/2

3
###
Let P be the point on the parabola, y^2 = 8x

Let P be the point on the parabola, y^{2} = 8x which is at a minimum distance from the centre C of the circle, x^{2} + (y + 6)^{2} = 1. Then the equation of the circle, passing through C and having its centre at P is:

- x
^{2}+ y^{2}– x/4 + 2y – 24 = 0 - x
^{2}+ y^{2}– 4x + 8y + 12 = 0 - x
^{2}+ y^{2}– 4x + 9y + 18 = 0 - x
^{2}+ y^{2}– x + 4y – 12 = 0

4
###
The locus of the foot of perpendicular drawn from the center

The locus of the foot of perpendicular drawn from the center of the ellipse x^{2} + 3y^{2} = 6 on any tangent to it is

- (x
^{2}- y^{2})^{2}= 6x^{2}+ 2y^{2} - (x
^{2}- y^{2})^{2}= 6x^{2}- 2y^{2} - (x
^{2}+ y^{2})^{2}= 6x^{2}- 2y^{2} - (x
^{2}+ y^{2})^{2}= 6x^{2}+ 2y^{2}

5
###
An ellipse is drawn by taking a diameter of the circle (x - 1) + y = 1

An ellipse is drawn by taking a diameter of the circle (x - 1) + y = 1 as its semi-minor axis and a diameter of the circle x^{2} + (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

- x
^{2}+ 4y^{2}= 8 - 4x
^{2}+ y^{2}= 4 - x
^{2}+ 4y^{2}= 16 - 4x
^{2}+ y^{2}= 8

6
###
The slope of the line touching both the parabolas

The slope of the line touching both the parabolas y^{2} = 4x and x^{2} = -32y is

- 1/2
- 2/3
- 3/2
- 1/8

7
###
The eccentricity of the hyperbola whose length of the latus rectum

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal half of the distance between its foci, is:

- 4/√3
- √3
- 4/3
- 2/√3

8
###
Let O be the vertex and Q be any point on the parabola, x^2 = 8y

Let O be the vertex and Q be any point on the parabola, x^{2} = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is

- x
^{2}= y - y
^{2}= x - x
^{2}= 2y - y
^{2}= 2x

9
###
Let a, b, c and d be non-zero numbers

Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax + 2ay + c = 0 and 5bx + 2by + d = 0 lies in the fourth quadrant and is equidistant from the two axes then

- 3bc - 2ad = 0
- 3bc + 2ad = 0
- 2bc + 3ad = 0
- 2bc - 3ad = 0

10
###
If the pair of straight lines x^2 - 2pxy - y^2 = 0

If the pair of straight lines x^{2} - 2pxy - y^{2} = 0 and x^{2} - 2qxy - y^{2} = 0 be such that each pair bisects the angle between the other pair, then

- pq = -1
- p = -q
- pq = 1
- p = q

11
###
The shortest distance between the line y - x = 1 and curve

The shortest distance between the line y - x = 1 and curve y = x^{2} is

- √3/4
- 4/√3
- 8/3√2
- 3√2/8

12
###
If the line 2x + y = k passes through the point which divides

If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3:2, then k equals

- 5
- 6
- 29/5
- 11/5

13
###
The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is

- x
^{2}+ y^{2}+ 2x + 2y = 47 - x
^{2}+ y^{2}+ 2x - 2y = 47 - x
^{2}+ y^{2}- 2x + 2y = 62 - x
^{2}+ y^{2}- 2x + 2y = 47

14
###
The two circles x^2 + y^2 = ax and x^2 + y^2 = c^2

The two circles x^{2} + y^{2} = ax and x^{2} + y^{2} = c^{2} (c > 0) touch each other if

- a = 2c
- |a| = 2c
- |a| = c
- 2|a| = c

15
###
The locus of the centre of a circle which touches the circle

The locus of the centre of a circle which touches the circle |z - z_{1}| = a and |z - z_{2}| = b externaly (z, z_{1} & z_{2} are complex numbers) will be

- a hyperbola
- an ellipse
- a circle
- a straight line

16
###
A triangle with vertices (4, 0), (-1, -1), (3, 5) is

A triangle with vertices (4, 0), (-1, -1), (3, 5) is

- right angled but not isosceles
- neither right angled nor isosceles
- isosceles and right angled
- isosceles but not right angled

17
###
If the two circles (x-1)^2 + (y-3)^2 = r^2 and x^2 + y^2 - 8x + 2y + 8 = 0 intersect in two distinct point

If the two circles (x-1)^{2} + (y-3)^{2} = r^{2} and x^{2} + y^{2} - 8x + 2y + 8 = 0 intersect in two distinct point, then

- 2 < r < 8
- r = 2
- r > 2
- r < 2

18
###
Let P be the point on the parabola, y^2 = 8x which is at a minimum distance

Let P be the point on the parabola, y^{2} = 8x which is at a minimum distance from the centre C of the circle, x^{2} + (y + 6)^{2} = 1. Then the equation of the circle, passing through C and having its centre at P is:

- x
^{2}+ y^{2}– x + 4y – 12 = 0 - x
^{2}+ y^{2}– x/4 + 2y – 24 = 0 - x
^{2}+ y^{2}– 4x + 9y + 18 = 0 - x
^{2}+ y^{2}– 4x + 8y + 12 = 0

19
###
The eccentricity of the hyperbola whose length of the latus rectum is equal to 8

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal half of the distance between its foci, is:

- 4/√3
- 2/√3
- √3
- 4/3