Let the point A divide the line segment joining the points P(–1, –1, 2) and Q(5, 5,10) internally in the ratio r : 1 (r > 0). If O is the origin and $(\vec{OQ} \cdot \vec{OA}) - \frac{1}{5} | \vec{OP} \times \vec{OA} |^{2} = 10$ , then the value of r is :

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$\sqrt{7}$

answer is D.

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Detailed Solution

$\begin{array}{l} A \equiv (\frac{5 r - 1}{r + 1}, \frac{5 r - 1}{r + 1}, \frac{10 r + 2}{r + 1}) \\ (\vec{OQ} \cdot \vec{OA}) - \frac{| \vec{OP} \times \vec{OA} |^{2}}{5} = 10 . . . (1) \\ \vec{OQ} \cdot \vec{OA} = \frac{5}{r + 1} (30 r + 2) \end{array}$
$| \vec{OP} \times \vec{OA} |^{2} = \frac{r^{2}}{(r + 1)^{2}} (800)$
so by equation (1)
$\begin{array}{l} \frac{10}{r + 1} (15 r + 1) - \frac{1}{5} \frac{r^{2} (800)}{(r + 1)^{2}} = 10 \\ 2 r^{2} - 14 r = 0 \\ r = 7, r \neq 0 \end{array}$