Question

If the two lines $\frac{x-1}{2}=\frac{1-y}{-a}=\frac{z}{4}$ and $\frac{x-3}{1}=\frac{2y-3}{4}=\frac{z-2}{2}$ are perpendicular, then the value of $a$ is equal to

• A. $-4$
• B. $5$
• C. $-5$
• D. $4$
• E. $-2$

Answer 1

Answer: (C)

$-5$

Given equation of lines

$\frac{x-1}{2}=\frac{1-y}{-a}=\frac{z}{4}$
$\Rightarrow \frac{x-1}{2}=\frac{y-1}{a}=\frac{z-0}{4}...\left(i\right)$
and $\frac{x-3}{1}=\frac{y-\frac{3}{2}}{2}=\frac{z-2}{2}...\left(ii\right)$

Here $a_1=2,b_1=a,c_1=4;a_2=1,b_2=2,c_2=2$

Since lines (i) and (ii) are perpendicular
$\therefore a_1{a}_2+b_1{b}_2+c_1{c}_2=0$

$\Rightarrow 2\times 1\times +a\times 2+4\times 2=0$

$\Rightarrow 2a=-10\Rightarrow a=-5$