Distance Formula

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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 (x , y) be cartesian co-ordinate of point P.

then,    OM = x , PM = y

OP = r  and ∠POM = θ₁

now ,  x = r.cosθ₁ and y₁ = r₁.sinθ₁

similarly,  (x , y) be cartesian co-ordinate of point Q.

then,    ON = x , QN = y

OQ = r  and ∠QON = θ

now ,  x = r.cosθ  and y = r.sinθ To calculate θ1 and θ2 :

we know that ,

x₁= r₁cosθ₁  and y₁ = r₁sinθ₁

tanθ₁ = y₁ / x₁

similarly ,

x = rcosθ  and y = rsinθ

tanθ = y / x 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(x,y) and Q(x,y) divides a line ax + by + c = 0 in the ratio :

-(ax +by + c)/(ax +by + 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 –