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 the coordinates of P be $(x, y, z)$. Since P divides the segment joining $(3, 6, -1)$ and $(6, 2, -2)$, we use the section formula to find the coordinates of P. The $y$-coordinate of P is 4, so the ratio of division is: \[ \frac{y - 6}{2 - y} = \frac{4 - 6}{2 - 4} = 1 \] Thus, the ratio of division is 1:1, and P divides the line segment equally. The $z$-coordinate of P is the average of the $z$-coordinates of the endpoints: \[ z = \frac{-1 + (-2)}{2} = -\frac{3}{2} \] Thus, the correct $z$-coordinate is $\frac{3}{2}$.