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

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

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

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

(x² - y²)² = 6x² + 2y²
(x² - y²)² = 6x² - 2y²
(x² + y²)² = 6x² - 2y²
(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

x² + 4y² = 8
4x² + y² = 4
x² + 4y² = 16
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

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

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

x² = y
y² = x
x² = 2y
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

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

pq = -1
p = -q
pq = 1
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

√3/4
4/√3
8/3√2
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

5
6
29/5
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

x² + y² + 2x + 2y = 47
x² + y² + 2x - 2y = 47
x² + y² - 2x + 2y = 62
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

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

a hyperbola
an ellipse
a circle
a straight line

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

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

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

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

x² + y² – x + 4y – 12 = 0
x² + y² – x/4 + 2y – 24 = 0
x² + y² – 4x + 9y + 18 = 0
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:

4/√3
2/√3
√3
4/3