Let be an integer.

Question

Let $\lambda$ be an integer. If the shortest distance between the lines $x-\lambda=2y-1=-2z$ and $x=y+2\lambda=z-\lambda$ is $\dfrac{\sqrt7}{2\sqrt2}$, then the value of $|\lambda|$ is:

Answer: 1

Accepted Answer

Line 1: $\dfrac{x-\lambda}{2}=\dfrac{y-1/2}{1}=\dfrac{z}{-1}$, point $\left(\lambda,\tfrac12,0\right)$.

Line 2: $\dfrac{x}{1}=\dfrac{y+2\lambda}{1}=\dfrac{z-\lambda}{1}$, point $(0,-2\lambda,\lambda)$.

Computing the shortest-distance formula:
$$\frac{\left|-5\lambda-\frac32\right|}{\sqrt{14}}=\frac{\sqrt7}{2\sqrt2} \implies |10\lambda+3|=7.$$
$$10\lambda=4\ \text{or}\ -10 \implies \lambda=\frac25\ \text{or}\ -1.$$
Since $\lambda$ is an integer: $\lambda=-1 \implies |\lambda|=1.$

Solutions · 1