Content-
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 ✍︎ :
- 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
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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.
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