Question

In the xy-plane, three distinct lines $l_1,l_2,l_3$ are concurrent at $M(\lambda ,0)$. Also the lines $l_1,l_2,l_3$ are normals to the parabola $y^2=6x$ at the points $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ respectively. Then

• A. $\lambda <-5$
• B. $\lambda >3$
• C. $-5<\lambda <-3$
• D. $0<\lambda <3$

Answer 1

The correct option is B $\lambda >3$

Any normal is $y=mx-2am-a{m}^3$, where $a=\frac{3}{2}$
$\therefore (\lambda ,0)\Rightarrow 0=m\lambda -2am-a{m}^3$
$\Rightarrow m=0,m^2=\frac{\lambda }{a}-2>0$
$\Rightarrow \lambda >2a\Rightarrow \lambda >3$