Question

The lines $a{x}^2+2hxy+b{y}^2=0$ are equally inclined to the lines

$a{x}^2+2hxy+b{y}^2+\lambda (x^2+y^2)=0$ for

• A. = 1 only
• B. = 2 only
• C. for any value of
• D. None of these

Answer 1

The correct option is C

for any value of

Equation of the bisectors of the angle between the lines

$a{x}^2+2hxy+b{y}^2+\lambda (x^2+y^2)=0$

$\frac{x^2-y^2}{(a+\lambda )-(b-\lambda )}=\frac{xy}{h}$

$\frac{x^2-y^2}{a-b}=\frac{xy}{h}$

Which is same as the equation of the bisectors of the angles between the lines $a{x}^2+2hxy+b{y}^2=0$

Equation of the bisectors of the angle between these lines doesn't depend on $\lambda$.

Thus, the two lines pairs are equally inclined to each other for any value of $\lambda$

Hence correct option is C.