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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