Show that the equation of the pair of lines bisecting the angles between the pair of bisectors of the angles between the pair of lines `a x^2+2h x y+b y^2=0` is `(a-b)(x^2-y^2)+4h x y=0.`

Explanation

Equation of the given lines is `ax^(2)+2hxy+by^(2)=0`. Equation of the pair of bisectors is
`h(x^(2)-y^(2))=(a-b)xy`
or `hx^(2)-(a-b)xy-hy^(2)=0` (1)
`:. A=h,B=-h,2H=-(a-b)`
Equation of the pair of bisectors of (1) is
`H(x^(2)-y^(2))=(A=B)xy`
or `-(a-b)/(2)(x^(2)-y^(2))=2hxy`
or `-(a-b)(x^(2)-y^(2))=4hxy`
or ` (a-b)(x^(2)-y^(2))+4hxy=0`