Question

If P is a point on the line segment joining $(3, 6, -1)$ and $(6, 2, -2)$ and the $y$-coordinate of P is 4, then its $z$-coordinate is :

• A. $-\frac{3}{2}$
• B. 0
• C. 1
• D. $\frac{3}{2}$

Hint:

For dividing line segments, use the section formula and find the ratio based on the known coordinates of the point.

Answer 1

The Correct Option is D

Let P have coordinates $(x, y, z)$. P divides the segment connecting $(3, 6, -1)$ and $(6, 2, -2)$. Using the section formula: The $y$-coordinate of P is 4. The ratio of division is determined by: \[ \frac{y - 6}{2 - y} = \frac{4 - 6}{2 - 4} = 1 \] This indicates a 1:1 ratio, meaning P bisects the segment. The $z$-coordinate of P is the average of the endpoints' $z$-coordinates: \[ z = \frac{-1 + (-2)}{2} = -\frac{3}{2} \] Therefore, the correct $z$-coordinate is $\frac{3}{2}$.