If the distance from point $P (1, 1, 1)$ to the line passing through the points $Q (0, 6, 8)$ and $R (- 1, 4, 7)$ is expressed in the form $\sqrt{\frac{p}{q}}$ where p and q are coprime, then the value $\frac{(p + q) (p + q - 1)}{2} =__$

see full answer

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

answer is 4950.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Equation of RQ is $\frac{x - 0}{0 - (- 1)} = \frac{y - 6}{6 - 4} = \frac{z - 8}{8 - 7}$

$\Rightarrow \frac{x}{1} = \frac{y - 6}{2} = \frac{z - 8}{1} = λ$

$1 (λ - 1) + 2 (2 λ + 6 - 1) + 1 \cdot (λ + 8 - 1) = 0$

$\Rightarrow 6 λ + 16 = 0 \Rightarrow λ = \frac{- 8}{3}$

$\Rightarrow M (\frac{- 8}{3}, \frac{2}{3}, \frac{16}{3})$

$PM = \sqrt{{(\frac{11}{3})}^{2} + {(\frac{1}{3})}^{2} + {(\frac{13}{3})}^{2}} = \sqrt{\frac{291}{9}} = \sqrt{\frac{97}{3}}$

$\frac{(p + q) (p + q - 1)}{2} = \frac{(97 + 3) (97 + 3 - 1)}{2} = 4950$

Watch 3-min video & get full concept clarity