Let point A divide the line segment joining P(-1,-1,2) and Q(5,5,10) internally in the rat…

Question

Let point $A$ divide the line segment joining $P(-1,-1,2)$ and $Q(5,5,10)$ internally in the ratio $r:1$ ($r>0$). If $O$ is the origin and $(\vec{OQ}\cdot\vec{OA}) - \dfrac15|\vec{OP}	imes\vec{OA}|^2 = 10$, then the value of $r$ is:

Accepted Answer

$A = \left(\dfrac{5r-1}{r+1},\dfrac{5r-1}{r+1},\dfrac{10r+2}{r+1}\right)$.

Computing $\vec{OQ}\cdot\vec{OA} = \dfrac{10}{r+1}(15r+1)$ and $|\vec{OP}\times\vec{OA}|^2 = \dfrac{800\,r^2}{(r+1)^2}$, substituting into the given equation:
$$\frac{10}{r+1}(15r+1) - \frac15\cdot\frac{800r^2}{(r+1)^2} = 10$$

This simplifies to $2r^2-14r=0 \Rightarrow r(r-7)=0$. Since $r>0$, $r=7$.

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