Three Dimensional Geometry Question 357
Question: If OABC is a tetrahedron where O is the origin and A, B, C are three other vertices with position vectors $ \overset{\to }{\mathop{a}},,\overset{\to }{\mathop{b}}, $ and $ \overset{\to }{\mathop{c}}, $ respectively, then the centre of sphere circumscribing the tetrahedron is given by the position vector
Options:
A) $ \frac{a^{2}(\vec{b}\times \vec{c})+b^{2}(\vec{c}\times \vec{a})+c^{2}(\vec{a}\times \vec{b})}{2[\vec{a},\vec{b},\vec{c}]} $
B) $ \frac{b^{2}(\vec{b}\times \vec{c})+a^{2}(\vec{c}\times \vec{a})+c^{2}(\vec{a}\times \vec{b})}{[\vec{a},\vec{b},\vec{c}]} $
C) $ \frac{b^{2}(\vec{b}\times \vec{c})+a^{2}(\vec{c}\times \vec{a})+c^{2}(\vec{a}\times \vec{b})}{2[\vec{a},\vec{b},\vec{c}]} $
D) $ \frac{a^{2}(\vec{a}\times \vec{b})+b^{2}(\vec{b}\times \hat{c})+c^{2}(\vec{c}\times \vec{a})}{2[\vec{a},\vec{b},\vec{c}]} $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] If the centre $ ‘P’ $ is with position vector $ \vec{r}, $ Then $ \vec{a}-\vec{r}=\overrightarrow{PA},\vec{b}-\vec{r}=\overrightarrow{PB},\vec{c}-\vec{r}=\overrightarrow{PC,} $ Where $ | \overrightarrow{PA} |=| \overrightarrow{PB} | $ $ =| \overrightarrow{PC} |=| \overrightarrow{OP} |=| {\vec{r}} | $ Consider $ | \vec{a}-\vec{r} |=| {\vec{r}} | $
$ \Rightarrow (\vec{a}-\vec{r}).(\vec{a}-\vec{r})=\vec{r}.\vec{r} $
$ \Rightarrow a^{2}=-2\vec{a}.\vec{r}+r^{2}=r^{2} $
$ \Rightarrow a^{2}=2\vec{a}.\vec{r} $ Similarly, $ b^{2}=2\vec{b}.\vec{r} $ and $ c^{2} $ $ =2\vec{c}.\vec{r} $ Since, $ (\vec{b}\times \vec{c}),(\vec{c}\times \vec{a}) $ and $ (\vec{a}\times \vec{b}) $ are non-coplanar, then $ \vec{r}=x(\vec{b}\times \vec{c})+y(\vec{c}\times \vec{a})+z(\vec{a}\times \vec{b}) $ $ \vec{a}.,\vec{r}=x,\vec{a}.(\vec{b}\times c)+y.0+z.0=x[\vec{a},\vec{b},\vec{c}] $
$ \Rightarrow x=\frac{\vec{a}.\vec{r}}{[\vec{a},\vec{b},\vec{c}]}=\frac{a^{2}}{2[\vec{a},\vec{b},\vec{c}]} $ Similarly, $ y=\frac{b^{2}}{2[\vec{a},\vec{b},\vec{c}]} $ and $ z=\frac{c^{2}}{2[\vec{a},\vec{b},\vec{c}]} $ Therefore, $ \vec{r} $ $ =\frac{a^{2}(\vec{b}\times \vec{c})+b^{2}(\vec{c}\times \vec{a})+c^{2}(\vec{a}\times \vec{b})}{2[\vec{a},\vec{b},\vec{c}]} $
BETA
