First slide

Straight lines in 3D

Question

If $P (4, - 1, 3), Q (0, 6, 7), R (- 2, 1, 9)$ are vertices of a triangle $P Q R$ ,the equation of the median through $R$ is $\frac{x - i}{m} = \frac{y - c}{s} = \frac{z - t}{y}$ , then $[m + y + s + t + i + c] =$ _____ .(where[.] denotes greatest integer function) and $m, s, y$ are numerically least possible integers.

Moderate

A

7
B

12
C

9
D

16

Solution

Midpoint of the side $P Q$ is $S = (\frac{4 + 0}{2}, \frac{- 1 + 6}{2}, \frac{3 + 7}{2}) = (2, \frac{5}{2}, 5)$
The direction ratios of the median $R S$ are $〈2 + 2, \frac{5}{2} - 1, 5 - 9〉 = 〈4, \frac{3}{2}, - 4〉 = 〈8, 3, - 8〉$
Hence, the equation of the median is $\frac{x - 2}{8} = \frac{y - \frac{5}{2}}{3} = \frac{z - 5}{- 8}$
Therefore, $[m + y + s + t + i + c] = [2 + \frac{5}{2} + 5 + 8 + 3 - 8] = [12.5] = 12$

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