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

[JEE Main 2025, 23 Jan (Shift 2)]

Question

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

[JEE Main 2025, 23 Jan (Shift 2)]

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Let the point A divide the line segment joining the points \(\mathrm{P}(-1,-1,2)\) and \(\mathrm{Q}(5,5,10)\) internally…

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