The position vectors of the vertices of a triangle are $3\hat{i}+4\hat{j}+5\hat{k},\hat{i}+7\hat{k}$ and $5\hat{i}+5\hat{j}$ . The distance between the circumcentre and orthocenter is $\sqrt{k}$ . Then k is equal to_____

(306.00)

Let $a=3\hat{i}+4\hat{j}+5\hat{k},b=\hat{i}+7\hat{k}$

And $c=5\hat{i}+5\hat{j}$

Then $\left|a\right|=\left|b\right|=\left|c\right|$

So, origin is the circumcentre and the orthocenter is

$a+b+c=9\hat{i}+9\hat{j}+12\hat{k}$
The distance between the circumcentre and orthocenter is
$|a+b+c|=\sqrt{9^2+9^2+{12}^2}=\sqrt{306}$
So, k = 306