The number of values of `lambda` for which the bisectors of the angle between the lines `ax^2 + 2hxy + by^2 + lambda(x^2 + y^2) = 0` are the same as those of `ax^2+2hxy+by^2=0` is

Options

A. a
B. b
C. h
D. any real number

Answer

D

Explanation

We have , `ax^2+2hxy+by^2=0` ...(i)
`ax^2+2hxy+by^2+lambda(x^2+y^2)=0`.....(ii)
Equation of bisector of tha angle between the lines represented by Eq.(i) is `(x^2-y^2)/(a-b)=(xy)/(h)`
`rArr h(x^2-y^2)=(a-b)xy`
`rArr hx^2-(a-b)xy-hy^2=0`.....(iii)
and equation of bisector of the angle between the lines represented by Eq.(ii) is `(x^2-y^2)/(a+lambda-b-lambda)=(xy)/(h)`
`rArr h(x^2-y^2)=(a-b)xy`
`rArr hx^2-(a-b)xy-hy^2=0` ....(iv)
Eqs. (iii) and (iv) are coincident for any value of `lambda` because it does not depend on `lambda`.