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 $l_1 x + m_1y+n_1 = 0$ and $l_2 x + m_2y +n_2 = 0$ are conjugate w.r.t. parabola $y^2 = 4ax$, then
$l_1 n_2 +l_2n_1= 2am_1m_2\quad ...(1)$
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 $
$\Rightarrow \lambda = - 3$.