If the lines $\frac{x-5}{5λ+2}=\frac{2-y}{5}=\frac{1-z}{-1}$ and $x=\frac{y+1/2}{2λ}=\frac{z-1}{3}$ are perpendicular, then the value of $λ$ is equal to

1

The correct answer is Option (2) → 1

$\text{Line 1: }\frac{x-5}{5\lambda+2}=\frac{2-y}{5}=\frac{1-z}{-1}\;\Rightarrow\;\text{DR } \langle\,5\lambda+2,\,-5,\,1\,\rangle$

$\text{Line 2: }x=\frac{y+\frac{1}{2}}{2\lambda}=\frac{z-1}{3}\;\Rightarrow\;\text{DR } \langle\,1,\,2\lambda,\,3\,\rangle$

$\text{For perpendicular lines: dot product }=0$

$(5\lambda+2)(1)+(-5)(2\lambda)+(1)(3)=0$

$5\lambda+2-10\lambda+3=0\;\Rightarrow\;-5\lambda+5=0\;\Rightarrow\;\lambda=1$

$\lambda=1$