Co-ordinate Geometry

The distance between points A(x₁, y₁) and B(x₂, y₂) is given by,

1. Given Points: A(2, 3) and B(4, 1)
we can see that,
x₁ = 2
x₂ = 4
y₁ = 3
y₂ = 1
Putting the values in the distance formula we get,
d =

⇒ d =

⇒ d = √8

2. Given Points: P(-5, 7) and Q(-1, 3)
we can see that,
x₁ = -5
x₂ = -1
y₁ = 7
y₂ = 3
Putting these values in distance formula we get,

d =

d = √32

3. Given Points: R(0, -3), S(0, 5/2)
we can see that,
x₁ = 0
x₂ = 0
y₁ = -3
y₂ = 5/2

d =

d =

d =

4. Given Points: L(5, -8), M(-7, -3)

we can see that,
x₁ = 5
x₂ = -7
y₁ = -8
y₂ = -3

On putting these values in distance formula we get,

d =

d =

d = √169 = 13

5. Given Points: T(-3, 6), R(9, -10)

we can see that,
x₁ = -3
x₂ = 9
y₁ = 6
y₂ = -10

On putting these values in distance formula we get,

d =

d =

d = 20

6. Given Points: W(), X(11, 4)

we can see that,
x₁ = -7/2
x₂ = 11
y₁ = 4
y₂ = 4

On putting these values in distance formula we get,

d =

d =

d =

If Three points (a,b), (c,d), (e,f) are collinear then the area formed by the triangle by the three points is zero.

Area of a triangle = ...(1)

1.
(a,b) = (1,-3)

(c,d) = (2,-5)

(e,f) = (-4,7)

Area =

Area = = 0

Hence the points are collinear.

2.
(a,b) = (-2,3)

(c,d) = (1,-3)

(e,f) = (5,4)

Area =

Area =

Hence the points are not collinear.

3.
(a,b) = (0,3)

(c,d) = (2,1)

(e,f) = (3,-1)

Area =

Area =

Hence the points are non collinear.

4.
(a,b) = (-2,3)

(c,d) = (1,2)

(e,f) = (4,1)

Area =

Area =

Hence the points are collinear.

A point in the x = axis is of the form (a,0)

Distance d between two points(a,b) and (c,d)is given by

Distance between (-3,4) and (a,0) =

D =

D

Distance between (1,-4) and (a,0)

D =

D =

As the two points are equidistant from the point (a.0)

=

Squaring both sides, we get

(1-a)² + 16 = (3 + a)² + 16

1 + a² -2a = 9 + a² + 6a

8a = -8

a = -1

Hence the point is (-1,0)

In a right angles triangle ABC, right angled at B, according to the pythagoras theorem

AB² + BC² = AC²

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

For the given points Distance between P and Q is

PQ = =

QR = =

PR = = =

PQ² = 16

QR² = 25

PR² = 41

As PQ² + QR² = PR²

Hence the given points form a right angled triangle.

In a parallelogram, opposite sides are equal and parallel.

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

For the given points, length PQ =

PQ =

Length QR =

QR = =

Length RS =

RS = =

Length SP =

SP = =

As PQ = RS and QR = SP

Checking for slopes

Slope of a line between two points (a,b) and (c,d) is

Slope PQ = = 1

Slope QR = =

Slope RS =

Slope SP =

As PQ = RS and their slope = 1

And

QR = SP and their slope = -1.

Hence the given points form a parallelogram.

In a Rhombus the sides are equal and the diagonals bisect each other at 90°

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Length AB = =

Length BC = =

Length CD = =

Length AD = =

Slope of a line between two points (a,b) and (c,d) is

Slope of Diagonal AC = = 1

Slope of diagonal BD = = -1

Note: If the Product of slopes of two lines = -1 then they are perpendicular to each other.

As the product of slopes pf two diagonals = -1. Hence they're perpendicular to each other.

Hence The given points form a rhombus.

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Distance between LM = = 10

Squaring both sides, we get

(x-1)² + 64 = 100

(x-1)² = 36

x-1 = ±6

Hence x = 7 or -5

For an equilateral triangle, all its sides are equal.

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Length AB = = = 4

Length BC = = = 4

Length AC = = = 4

Hence The given points form an equilateral triangle.

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

In the given question

x = = 1

y =

y = 3

Hence the coordinates of the point are (1,3).

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and


Where m and n is defined as the ratio in which the line segments are divided

1. x =

x =

y =

y =

2. x =

X =

y =

y =

3. x = = 0

y =

y =

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

In the given question,

Let the point T divide the line PQ in the ratio m:n

Here x = -1 and y = 6

...(1)

.... (2)

Simplifying (1) we get,

-m-n = -3n + 6m

2n = 7m

Simplifying (2) we get,

6m + 6n = 10n-8m

14m = 4n

From both we get

Hence the point T divides PQ in the ratio 2:7

According to the mid point theorem the coordinates of the point P(x,y) dividing the line formed by A(a,b) and B(c,d) is given by:

In the given question A = (1,-3) and midpoint P is (-2,0).

Let coordinates of B be (c,d)

Then,

And

Solving for c and d, we get

-4 = 2 + c

c = -6

d = 3

Hence the coordinates of point B are (-6,3).

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

In the given question,

Let the point P divide AB is the ratio 1:k

Y coordinate of P

Simplifying

7k + 7 = 9k + 2

2k = 5

k =

and the ratio = 1:

=

Therefore point P divides AB in the ratio 2:5

According to the mid point theorem the coordinates of the point P(x,y) dividing the line formed by A(a,b) and B(c,d) is given by:

The coordinates of midpoint(x,y) are

Hence the coordinates are (11,18)

The coordinates of the centroid (x,y) od a triangle formed by points (a,b), (c,d), (e,f) is given by

1.

2.

3.

The coordinates of the centroid (x,y) od a triangle formed by points (a,b), (c,d), (e,f) is given by

In the given question (x,y) = (-4,-7)

Hence

Solving for e, we get

e = -1

Solving for f, we get

f = -7

Hence the coordinates of the third point are (-1,-7)

The coordinates of the centroid (x,y) od a triangle formed by points (a,b), (c,d), (e,f) is given by

In the given question:

Solving for h we get

h = 7

Solving for k we get

k = 18

let The points of trisection of a given line AB be P and Q

Then the ratio AP:PQ:QB = 1:1:1

Hence we get AP:PB = 1:2

And AQ:QB = 2:1

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

To find point P(x,y)

x = 0

y = 2

To find the point Q(x',y')

x' = -2

y' = -3

Hence point P = (0,2) and Q = (-2,-3)

let the points dividing AB be C,D,E.

AC:CD:DE:EB∷1:1:1:1

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

For C m:n ∷ 1:3

For D m:n ∷2:2

For E m:n ∷ 3:1

Hence coordinates of C = (-9,-8)

D = (-4,-6)

E = (1,-4)

Let the points dividing AB be C,D,E,F

AC:CD:DE:EF:FB∷1:1:1:1:1

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

For C m:n ∷ 1:4

For D m:n ∷ 2:3

For E m:n ∷ 3:2

= 8

For F m:n ∷ 4:1

slope is given as the tangent of the angle formed with the positive direction of x-axis

1. tan 45° = 1

2. tan60° =

3. tan 90° = cannot be determined.

Slope m of a line passing through two points A(a,b) and B(c,d) ig given by

1. A (2, 3), B (4, 7)

2. P (-3, 1), Q (5, -2)

3. C (5, -2), D (7, 3)

4. L (-2, -3), M (-6, -8)

5. E(-4, -2), F (6, 3)

6. T (0, -3), S (0, 4)

So, slope cannot be determined.

Three points are said to be collinear if they all lie in a straight line.
If Three points (x₁, y₁), (x₂, y₂), (x₃, y₃) are collinear then no triangle can be formed using three points and so the area formed by the triangle by the three points is zero.

Area of Triangle = 1/2 [x₁ (y₂ − y₃) + x₂ (y₃ − y₁) + x₃ (y₁ − y₂)] ...(1)

1. For triangle, A(-1, -1), B(0, 1), C(1, 3)
Area =
= 1/2 [ -1(-2) + 0(3+1) + 1(-1-1)]

= 1/2 [2 + 0 - 2]

= 1/2 [2-2]
= 0
Hence the points are collinear

2. For triangle, D(-2, -3), E(1, 0), F(2, 1)
Using 1,
Area = 1/2 [-2(0-1) + 1(1-(-3)) + 2(-3-0)]
= 1/2 [-2(-1) + 1(1+3) + 2(-3)]
= 1/2 [ 2 + 4 -6 ]
= 1/2 [ 6 - 6 ]
= 1/2 (0)
= 0
Hence the points are collinear.

3. For triangle, L(2, 5), M(3, 3), N(5, 1)
Using 1,

Area = 1/2 [2(3-1) + 3(1-5) + 5(5-3)]
= 1/2 [ 2(2) + 3(-4) + 5(2)]
= 1/2 [ 4 - 12 + 10 ]
= 1/2 [ 14 - 12 ]

= 1/2 (2)
= 1

Hence the points are not collinear.

4. For triangle, P(2, -5), Q(1, -3), R(-2, 3)
Using 1,
Area = = 0
Area = 1/2 [ 2(-3-3) + 1(3-(-5)) + (-2)(-5-(-3))]
= 1/2 [ 2(-6) + 1(3+5) - 2 (-5+3)]
= 1/2 [ -12 + 8 -2(-2)]
= 1/2 [-12 + 8 +4]
= 1/2 [ -12 + 12]
= 1/2 (0)

Hence the points are collinear.

5. For triangle, R(1, -4), S(-2, 2), T(-3, 4)
Area =

Hence the points are collinear.

6. For triangle, A(-4, 4), K(-2, 5/ 2 ), N(4, -2)
Area =

Hence the points are collinear.

Slope m of a line passing through two points A(a,b) and B(c,d) is given by

Slope of AB =

Slope of BC =

Slope of AC =

In a parallelogram, opposite sides are equal and parallel.

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Slope m of a line passing through two points A(a,b) and B(c,d) ig given by

In the question,

AB = =

BC = =

CD = =

AD = =

Slope of AB =

Slope of BC =

Slope of CD =

Slope of AD =

As AB = DC and BC = AD

And Slope AB = Slope CD

Slope BC = slope AD

Hence the given points form a parallelogram.

Slope m of a line passing through two points A(a,b) and B(c,d) ig given by

In the given question

Simplifying

6 = k + 1

K = 5

Slope m of a line passing through two points A(a,b) and B(c,d) ig given by

In the given question

Simplifying

7-7k = 7

k = 0

two lines are said to be parallel if their slopes are equal

If PQ||RS then their slopes must be equal

Slope of PQ =

Slope pf RS =

As their slopes are equal, we get

Simplifying

k-1 = 4

k = 5

To be parallel to y-axis, it's x coordinate should remain the same. i.e. 1 and y coordinate can change.

A. x coordinate has changed.

B. x coordinate has changed.

C. x coordinate has changed.

D. x coordinate is same.

Therefore the answer is D.

To be on the X-axis, it's y coordinate = 0

And to be on the right of the origin, its x coordinate must be positive.

A. y is 0 but x is negative

B. y is not 0

C. y is not 0

D. y is 0 and x is positive.

Therefore the answer is D

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

d = = 5

Therefore answer is C

Slope = tangent of angle formed with positive x-axis

Slope = tan30° =

Hence answer is C

If Three points (a,b), (c,d), (e,f) are collinear then the area formed by the triangle by the three points is zero.

Area of a triangle = ...(1)

1. area =

Hence the points are collinear.

2. area =

Hence the points are collinear

3. area =

Hence the points are not collinear

According to the mid point theorem the coordinates of the point P(x,y) dividing the line formed by A(a,b) and B(c,d) is given by:

In question

Hence mid point is (6,13)

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

On y axis x coordinate = 0

Let the y-axis divide AB is ratio 1:k.

For x coordinate

Solving for k we get


3k - 9 = 0

3 k = 9

k = 3

Hence the y axis divides the given point in the ratio 3:1

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

As the point is on the x = axis it is of the form (x,0)

Distance from point P = =

Distance from point Q = =

As the two points are equidistant from (x,0)

Squaring both sides

(2-x)² + 25 = (2 + x)² + 81

Expanding and simplifying

-4x + 25 = 4x + 81

8x = -56

x = -7

Hence the point is (-7,0)

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

i. d =

=

=

(ii) P(-6, -3), Q(-1, 9)

d =




= √169

= 13

(iii) R(-3a, a), S(a, -2a)

d =




= 5a

The circumcentre is equidistant from all the points of the triangle.

Let the coordinates of circumcentre be (x,y)

= ...i

And

...ii

Squaring and simplifying i , we get

2x-2y = -6

Squaring and simplifying ii , we get

2x + 10y = 6

Solving the above equations, we get

x =

y =

hence the coordinates of circumcircle is ()

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

1. LM = =

MN = =

NL = =

As sum of any two sides are greater than the third side,

The following points form a scalene triangle.

2. PQ = =

QR = =

RP = =

As PQ + QR < RP

The following points donot form a triangle.

3. AB = = 4

BC = = = 4

AC = = 4

As AB = BC = AC

The following points form an equilateral triangle.

Slope of a line between two points (a,b) and (c,d) is

Slope = =

Simplifying

K = 5

Slope of a line between two points (a,b) and (c,d) is

Slope of AB =

Slope of AC =

As slopes are equal, the two lines are parallel.

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Slope of a line between two points (a,b) and (c,d) is

Distance PQ = =

Distance QR = =

Distance RS = =

Distance SP = =

Slope PQ =

Slope QR =

Slope RS =

Slope SP =

As opposite sides are equal and parallel, the points from a parallelogram.

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Slope of a line between two points (a,b) and (c,d) is

Note: If the Product of slopes of two lines = -1 then they are perpendicular to each other.

PQ = =

QR = =

RS = =

SP = =

Slope PQ =

Slope QR =

Slope RS =

Slope SP =

As opposite sides are equal and parallel and perpendicular to each other, points form a rectangle.

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by .....(1)

According to the mid point theorem the coordinates of the point P(x,y) dividing the line formed by A(a,b) and B(c,d) is given by:

Mid point of AB x coordinate =

Y coordinate =

Mid point of BC x coordinate =

Y coordinate =

Mid point of AC x coordinate =

Y coordinate = = 3

Length of median through A is the distance between pt A and the mid point of BC

D_a = = 5

Length of median through B is the distance between pt B and the mid point of AC

D_b = = 2

Length of median through C is the distance between pt C and the mid point of AB

D_c = =

The coordinates of the centroid (x,y) od a triangle formed by points (a,b), (c,d), (e,f) is given by

X coordinate =

Y coordinate =

Hence coordinates are (1,3)

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Slope of a line between two points (a,b) and (c,d) is

Note: If the Product of slopes of two lines = -1 then they are perpendicular to each other.

AB = =

BC = =

CD = =

AD = =

Slope AB =

Slope BC =

Slope CD =

Slope AD =

As all sides are equal and ajdacent sides are perndicular. Given points form a square.

Let the circumcentre be (x,y)

As the circumcentre is equidistant from all the 3 points, we get

= .......i

And

....................ii

Squaring both sides of i and simplifying, we get

-2x-10y = -30

Squaring both sides of ii and simplifying, we get

10x + 2y = 46

Solving the above equations, we get

x =

y =

Radius of circumcircle is the distance between any point on the triangle and the circumcentre.

Radius = =

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

X coordinate = = 7

Y coordinate = = 3

Hence the point is (7,3)

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Slope of a line between two points (a,b) and (c,d) is

AB = =

BC = =

CD = =

AD = =

Slope AB =

Slope BC =

Slope CD =

Slope AD =

As opposite sides are equal and parallel, it forms a parallelogram.

A point P(x,y) divides the line formed by points (a,b) and (c,d) in the ratio of m:n, then the coordinates of the point P is given by

and

Coordinates of R as QR:RS ∷ 1:1

X =

Y = = 16

As RS:SB∷1:1

Coordinates of B

X = 0

And

Y = 20

As PQ:QR∷1:1

Coordinates of P

x = 16

and

y = 12

As AP:PQ∷1:1

Coodinates of A

x = 20

and

y = 10

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Let the centre be A(x,y)

As it passes through the given points, distance between centre and the points is the radius.

AP =

AQ =

AR =

As AP = AQ

Squaring both sides

(x-6)² + (y + 6)² = (x-3)² + (y + 7)²

Simplifying

12y-12x + 72 = 14y-6x + 58

2y + 6x-14 = 0... (a)

AP = AR

Squaring both sides and simplifying

6y-6x + 54 = 0 ...(b)

Solving for x and y using (a) and (b)

We get x = 3; y = -2

Hence centre is (3,-2)

According to the distance formula, the distance 'd' between two points (a,b) and (c,d) is given by
.....(1)

Slope of a line between two points (a,b) and (c,d) is

In the given question, for it to be a parallelogram AD = BC and slope AD = Slope BC

And

AB = DC and Slope AB = Slope CD

Let D be (x,y)

As AD = BC we get

= = 2 ....i

As AB = CD we get

= = 4 .....ii

As slope AD = Slope BC

....iii

As slope AB = Slope DC

.....iv

From iii we gey y = 6

Putting y = 6 in (i) we get

x = 3

and putting y = 6 in (ii) we get x = 7

Hence the possible coordinates of the point D are (7,6) and (3,6).

Slope of a line between two points (a,b) and (c,d) is

A quadrilateral ABCD has diagonals AC and BD

Slope AC =

Slope BD =