P is a point on the line segment joining the points (3, 2, −1) and (6, 2, −2) internally. If x coordinate of P is 5, then its z-coordinate is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → −5/3
Let P divide the segment in the ratio k : 1. By the section formula:
\( \displaystyle P=\left(\frac{6k+3}{k+1},\ 2,\ \frac{-2k-1}{k+1}\right) \)
The given x-coordinate fixes k:
\( \displaystyle \frac{6k+3}{k+1}=5 \)
\( \displaystyle 6k+3=5k+5 \)
\( \displaystyle k=2 \)
\( \displaystyle z=\frac{-2(2)-1}{2+1}=-\frac{5}{3} \)
Shortcut: The x-coordinate goes from 3 to 6, so x = 5 is \( \frac{2}{3} \) of the way. The z-coordinate goes from −1 to −2, so it is \( -1-\frac{2}{3}=-\frac{5}{3} \).