Question

If $2x+3y+12=0$ and $x-y+4\lambda$ = 0 are conjugate with respect to the parabola $y^2=8x,$ then $\lambda$ is equal to

• A. 2
• B. -2
• C. 3
• D. -3

Answer 1

Answer: (D)

Using the condition that if two lines $l_{1}x+m_{1}y+n_1=0$ and $l_{2}x+m_{2}y+n_2=0$ are conjugate w.r.t. parabola $y^2=4ax$, then
$l_1{n}_2+l_2{n}_1=2a{m}_1{m}_2...(1)$
Given conjugate lines are $2x+3y+12=0$ and $x-y+4\lambda =0$ and equation of parabola is $y^2=8x$
Here, $l_1=2,m_1=3,n_1=12;l_2=1,m_2=-1$,
$n_2=4\lambda$ and $a=2$
from $(1)2\times 4\lambda +1\times 12=2\times 2\times 3\times (-1)$
$8\lambda =-12-12$
$\Rightarrow \lambda =-3$.