Question

If $y = mx$be one of the bisectors of the angle between the lines $ax^{2} - 2hxy + by^{2} = 0$, then

• A. $h(1 + m^{2}) + m(a - b) = 0$
• B. $h(1 - m^{2}) + m(a + b) = 0$
• C. $h(1 - m^{2}) + m(a - b) = 0$
• D. $h(1 + m^{2}) + m(a + b) = 0$

Answer 1

Answer: (C)

$h(1 - m^{2}) + m(a - b) = 0$

Answer 2

Here equation of one bisector of angle is $y - mx = 0,$ therefore equation of second is $x + my = 0$.

Hence combined equation is $(x + my)(y - mx) = 0$

$\Rightarrow - mx^{2} - xy(m^{2} - 1) + my^{2} = 0$ .….(i)

Also equations of bisectors of $ax^{2} - 2hxy + by^{2} = 0$ is

$- hx^{2} - (a - b)xy + hy^{2} = 0$ .....(ii)

Hence (i) and (ii) are the same equations, therefore

$\frac{m}{h} = \frac{m^{2} - 1}{(a - b)} \Rightarrow h(m^{2} - 1) = m(a - b)$

$\Rightarrow m(a - b) + h(1 - m^{2}) = 0$.