Angle Between Two Lines

Given pair of lines of

(%7B%7Bx%7D%5E%7B2%7D%7D%2B%7B%7By%7D%5E%7B2%7D%7D)si%7B%7Bn%7D%5E%7B2%7D%7D%5Calpha%20%3D%7B%7B(x%5C%2C%5Ccos%20%5C%2C%5Ctheta%20-y%5C%2C%5Csin%20%5Ctheta%20)%7D%5E%7B2%7D%7D

%5CRightarrow

%7B%7Bx%7D%5E%7B2%7D%7D%5C%2C%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20%2B%7B%7By%7D%5E%7B2%7D%7D%5C%2C%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20%3D%7B%7Bx%7D%5E%7B2%7D%7D%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta

%2B%7B%7By%7D%5E%7B2%7D%7D%5C%2C%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20-2xy%5C%2C%5Csin%20%5Ctheta%20%5C%2C%5Ccos%20%5Ctheta

%5CRightarrow

%7B%7Bx%7D%5E%7B2%7D%7D(%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20)%2B%7B%7By%7D%5E%7B2%7D%7D(%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20)

%2B2xy%5C%2C%5C%2C%5Csin%20%5Ctheta%20%5C%2C%5Ccos%20%5Ctheta%20%3D0

%5CRightarrow

%7B%7Bx%7D%5E%7B2%7D%7D(%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20)%2B%7B%7By%7D%5E%7B2%7D%7D(%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20)

%2B2(%5Csin%20%5Ctheta%20%5C%2C%5Ccos%20%5Ctheta%20)%5C%2Cxy%3D0

On comparing with

a%7B%7Bx%7D%5E%7B2%7D%7D%2Bb%7B%7By%7D%5E%7B2%7D%7D%2B2hxy%3D0%2C

We get,

a%3D%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20%2C%5C%2Cb%3D%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta

and

h%3D%5Csin%20%5C%2C%5Ctheta%20%5C%2C%5Ccos%20%5Ctheta

Let

%5Ctheta

be the angle between the pair of lines.

%5Ctherefore

%5Ctan%20%5Ctheta%20%3D%5Cleft%7C%20%5Cfrac%7B2%5C%2C%5Csqrt%7B%7B%7Bh%7D%5E%7B2%7D%7D-ab%7D%7D%7Ba%2Bb%7D%20%5Cright%7C

%3D%5Cleft%7C%20%5Cfrac%7B2%5Csqrt%7B%5Cbegin%7Balign%7D%20%26%20%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5C%2C%5Ctheta%20%5C%2C%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20-(%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20)%20%5C%5C%20%26%20%5Ctimes%20(%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20)%20%5C%5C%20%5Cend%7Balign%7D%7D%7D%7B%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20%2B%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20-%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20)%7D%20%5Cright%7C

%3D%5Cleft%7C%20%5Cfrac%7B2%5Csqrt%7B%5Cbegin%7Balign%7D%20%26%20%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20%5C%2C%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20-%7B%7B(%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20)%7D%5E%7B2%7D%7D%2B%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20%5C%5C%20%26%20%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20%2B%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20%5C%2C%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20-%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20%5C%2C%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20%5C%5C%20%5Cend%7Balign%7D%7D%7D%7B-(-1-2%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20)%7D%20%5Cright%7C

%3D%5Cleft%7C%20%5Cfrac%7B2%5Csqrt%7B%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20(%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Ctheta%20%2B%7B%7B%5Ccos%20%7D%5E%7B2%7D%7D%5Ctheta%20)-%7B%7B(si%7B%7Bn%7D%5E%7B2%7D%7D%5Calpha%20)%7D%5E%7B2%7D%7D%7D%7D%7B-%5Ccos%20%5C%2C2%5C%2C%5C%2C%5Calpha%20%7D%20%5Cright%7C

%3D%5Cleft%7C%20%5Cfrac%7B2%5Csqrt%7B%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20(1-%7B%7B%5Csin%20%7D%5E%7B2%7D%7D%5Calpha%20)%7D%7D%7B-%5Ccos%20%5C%2C2%5Calpha%20%7D%20%5Cright%7C

%5CRightarrow

%5Ctan%20%5Ctheta%20%3D%5Cleft%7C%20%5Cfrac%7B%5Csin%20%5C%2C2%5Calpha%20%7D%7B%5Ccos%20%5C%2C2%5Calpha%20%7D%20%5Cright%7C%3D%5Ctan%20%5C%2C2%5Calpha

%5CRightarrow

%5Ctheta%20%3D2%5Calpha

Chapter Questions
1 MCQs