Lengths of two opposite edges of a tetrahedron are a and b . Shortest distance between these edges is d and the angle between them is $θ$ . Such that its volume is $\frac{m}{n} a b d s i n θ$ . Find m + n (Where $\frac{m}{n}$ is a rational number in simplest form)

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answer is 7.

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Detailed Solution

$\begin{array}{l} | \bar{A B} | = | \bar{β} - \bar{α} | = a \\ | \bar{O C} | = | \bar{r} | = b \end{array}$
Eq of line $\bar{O C} \Rightarrow \bar{r} = t \bar{r}$
Eq of line AB = $\bar{α} + μ (\bar{β} - \bar{α})$
Shortest distance between them= $\frac{(\bar{α} - 0) \cdot (\bar{β} - \bar{α}) \times \bar{r}}{| (\bar{β} - \bar{α}) \times \bar{r} |}$
d = $\frac{[\bar{α} \bar{β} \bar{γ}]}{a b \sin θ}$
dabsin $θ$ = $[\bar{α} \bar{β} \bar{γ}]$
v= $\frac{1}{6}$ abd sin $θ$
k = 6