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### Distance  Formula

In X-Y Plane , The distance formula between two points P(x1 , y1) & Q(x2 , y2)–

let extends both points and create lines  meet at intersection point R. so, here PQR form a right angled triangle.

using Pythagoras theorem in right angled triangle PQR Note :

If Distance is given then use always ( ±  ) sign , to calculate distance between two points.

### Distance between two points in Polar Co-ordinates :

In polar co-ordinate system —

let O = Pole

OX = initial line.

let P & Q be two polar points whose polar co-ordinates

( r1 , θ1 ) & ( r2 , θ2 ) respectively. In fig.

OP = r1 , OQ = r2

∠POX = θ1   and  ∠QOX = θ2.

then ∠POQ = θ1 – θ2.

according to figure – In triangle ▲POQ : using  cosine rule ### Note ✍︎ :

1. Using this formula – make sure θ1 and θ2 are in radian , if not then convert into radian.
###### Conversion of Polar co-ordinate into Cartesian co-ordinate :

In Cartesian co-ordinate —

let (x1 , y1) be cartesian co-ordinate of point P.

then,    OM = x1 , PM = y1

OP = r1  and ∠POM = θ1

now ,  x1 = r1.cosθ1 and y1 = r1.sinθ1

similarly,  (x2 , y2) be cartesian co-ordinate of point Q.

then,    ON = x2 , QN = y2

OQ = r2  and ∠QON = θ2

now ,  x2 = r2.cosθ2  and y2 = r2.sinθ2 To calculate θ1 and θ2 :

we know that ,

x1 = r1cosθ1  and y1 = r1sinθ1

tanθ1 = y1 / x1

similarly ,

x2 = r2cosθ2  and y2 = r2sinθ2

tanθ2 = y2 / x2 #### Section Formula :

Point R(x , y) which divides the joining of two points P(x1 , y1) and Q(x2 , y2) in the given ratio m1:m2 , where  ( m1 , m2 > 0)

a) R(x , y) divides the PQ segment internally in the ratio m1:m2 b) R(x , y) divides the PQ segment externally in the ratio m1:m2 ———————————————————————————————————————

Note 1. To find ratio , using k:1.

if k is positive , then divides internally.

if k is negative , then divides externally.

Note 2.  Two points P(x1,y1) and Q(x2,y2) divides a line ax + by + c = 0 in the ratio :

-(ax1 +by1 + c)/(ax2 +by2 + c)

if ratio is negative  : divide externally.

if ratio is positive  : divide internally.

#### Test Yourself

3
Created on By Science++

Quiz

1 / 5

Find Distance between two points A(2,7) and B(-2, 6) is

2 / 5

Convert into cartesian co-ordinate into polar co-ordinates:

cartesian co-ordinates : ( -2 , -2 )

find polar co-ordinates in ( r , θ ) form

3 / 5

Find Distance between ( 3,∏/2 ) and  ( 3, ∏/6 )

4 / 5

Find the nature of the triangle whose vertices are-  A(2,7), B(4,-1) and C( -2,6) is —

5 / 5

The triangle joining the points P(0, 0 ), Q(3, ∏/2 ) and ( 3, ∏/6 ) is –