Question

If the distance from the point $P(1,1,1)$ to the line passing through the points $Q(0,6,6)$ and $R(-1,4,7)$ is expressed in the form $\sqrt{p/q}$ where p and q are coprime, then the value of $\frac{(p+q)(p+q-1)}{2}$
equals

• A. $5000$
• B. $5050$
• C. $5150$
• D. none

Answer 1

The correct option is D none

$a=0+1=1$

$b=6-4=2$

$c=6-7=-1$

equation of line :-

$\frac{x-0}{1}=\frac{y-6}{2}=\frac{z-6}{-1}$

$L_1\equiv 2x=y-6=12-2z$

From the equation of line :-

$\overline{S}=(1,2,-1)$

$Q\equiv (0,6,6)$

Then, $PQ\equiv \{0-1,6-1,6-1\}=(-1,+5,+5)$

$PQ\times \overline{S}=\left|\begin{array}{ccc}i& j& k\\ -1& 5& 5\\ 1& 2& -1\end{array}\right|=\hat{i}(\frac{-5}{2}-5)-\hat{j}(\frac{1}{2}-\frac{5}{2})+\hat{k}(-1-\frac{5}{2})$

$=\hat{i}\left(\frac{-15}{2}\right)-\hat{j}(-2)+\hat{k}\left(\frac{-7}{2}\right)$

$=\frac{-15\hat{i}}{2}+2\hat{j}-\frac{7\hat{k}}{2}$

$\equiv (\frac{-15}{2},2,\frac{-7}{2})$

$d=\frac{|\overline{pq}\times \overline{s}|}{|\overline{s}|}=\frac{\sqrt{\frac{225}{4}+4+\frac{49}{4}}}{\sqrt{\frac{1}{4}+1+\frac{1}{4}}}$ $=\frac{\sqrt{\frac{290}{4}}}{\sqrt{\frac{6}{4}}}=\sqrt{\frac{145}{3}}=\sqrt{\frac{145}{3}}$

$p=145$

$q=3$

$\frac{(p+q)(p+q-1)}{2}=\frac{148\times 147}{2}$

$=10,878$

$\left(D\right)$ none is the answer.