# Coordinate Geometry

### 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

1. 2y - x = 2
2. 4x - 2y = 1
3. 4x + 2y = 7
4. x + 2y = 4

### 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

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

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

Let P be the point on the parabola, y2 = 8x which is at a minimum distance from the centre C of the circle, x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is:

1. x2 + y2 – x/4 + 2y – 24 = 0
2. x2 + y2 – 4x + 8y + 12 = 0
3. x2 + y2 – 4x + 9y + 18 = 0
4. x2 + y2 – x + 4y – 12 = 0

### 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 x2 + 3y2 = 6 on any tangent to it is

1. (x2 - y2)2 = 6x2 + 2y2
2. (x2 - y2)2 = 6x2 - 2y2
3. (x2 + y2)2 = 6x2 - 2y2
4. (x2 + y2)2 = 6x2 + 2y2

### 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 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

1. x2 + 4y2 = 8
2. 4x2 + y2 = 4
3. x2 + 4y2 = 16
4. 4x2 + y2 = 8

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

The slope of the line touching both the parabolas y2 = 4x and x2 = -32y is

1. 1/2
2. 2/3
3. 3/2
4. 1/8

### 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:

1. 4/√3
2. √3
3. 4/3
4. 2/√3

### 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, x2 = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is

1. x2 = y
2. y2 = x
3. x2 = 2y
4. y2 = 2x

### 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

1. 3bc - 2ad = 0
2. 3bc + 2ad = 0
3. 2bc + 3ad = 0
4. 2bc - 3ad = 0

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

If the pair of straight lines x2 - 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 be such that each pair bisects the angle between the other pair, then

1. pq = -1
2. p = -q
3. pq = 1
4. p = q

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

The shortest distance between the line y - x = 1 and curve y = x2 is

1. √3/4
2. 4/√3
3. 8/3√2
4. 3√2/8

### 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

1. 5
2. 6
3. 29/5
4. 11/5

### 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

1. x2 + y2 + 2x + 2y = 47
2. x2 + y2 + 2x - 2y = 47
3. x2 + y2 - 2x + 2y = 62
4. x2 + y2 - 2x + 2y = 47

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

The two circles x2 + y2 = ax and x2 + y2 = c2 (c > 0) touch each other if

1. a = 2c
2. |a| = 2c
3. |a| = c
4. 2|a| = c

### 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 - z1| = a and |z - z2| = b externaly (z, z1 & z2 are complex numbers) will be

1. a hyperbola
2. an ellipse
3. a circle
4. a straight line

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

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

1. right angled but not isosceles
2. neither right angled nor isosceles
3. isosceles and right angled
4. isosceles but not right angled

### 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 = r2 and x2 + y2 - 8x + 2y + 8 = 0 intersect in two distinct point, then

1. 2 < r < 8
2. r = 2
3. r > 2
4. r < 2

### 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, y2 = 8x which is at a minimum distance from the centre C of the circle, x2 + (y + 6)2 = 1. Then the equation of the circle, passing through C and having its centre at P is:

1. x2 + y2 – x + 4y – 12 = 0
2. x2 + y2 – x/4 + 2y – 24 = 0
3. x2 + y2 – 4x + 9y + 18 = 0
4. x2 + y2 – 4x + 8y + 12 = 0

### 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:

1. 4/√3
2. 2/√3
3. √3
4. 4/3