Question

The circumcentre of the triangle formed by the points (1,2,3), (3,-1,5), (4,0,-3) is

• A. (1,1,1)
• B. $\left(\frac{7}{2},\frac{1}{2},1\right)$
• C. (3,3,3)
• D. $\left(\frac{7}{2},\frac{-1}{2},1\right)$

Answer 1

Answer: (D)

$\left(\frac{7}{2},\frac{-1}{2},1\right)$

We have

A $(1,2,3)$

B $(3,-1,5)$

C $(4,0,-3)$

Direction ratio of $\overrightarrow{AB}=(3-1,-1-2,5-3)=(2,-3,2)$

Direction ratio of $\overrightarrow{AC}=(4-1,0-2,-3-3)=(3,-2,-6)$

Now let us find dot product

$\overrightarrow{AB}\cdot \overrightarrow{AC}=(2,-3,2)(3,-2,-6)$

$=6+6-12=0$

As dot product of AB and $\overrightarrow{AC}$ is $0,$ hence

Triangle is right angled at A.

For right angled triangle, circumcentre is mid point of hypotenuse.

Hence Circumcentre $=$ Mid point if $BC$

$=(\frac{3+4}{2},\frac{-1+0}{2},\frac{5-3}{2})$

$=(\frac{7}{2},-\frac{1}{2},1)[$ Answer $]$