Question

If a point \( P \) on the line segment joining the points \( (3, 5, -1) \) and \( (6, 3, -2) \) has its \( y \)-coordinate 2, then its \( z \)-coordinate is

• A. \( \frac{2}{15} \)
• B. \( \frac{17}{3} \)
• C. \( \frac{15}{2} \)
• D. \( \frac{3}{17} \)

Hint:

To find the coordinates of a point on a line segment, use the parametric form of the line and solve for the unknown coordinates.

Answer 1

The Correct Option is C

Step 1: Find the parametric equations for the line.
The parametric equations of the line joining two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) are: \[ x = x_1 + t(x_2 - x_1), \quad y = y_1 + t(y_2 - y_1), \quad z = z_1 + t(z_2 - z_1) \] Substitute the given points into these equations.
Step 2: Solve for the parameter \( t \).
We know that the \( y \)-coordinate of the point is 2, so solve for \( t \) from the equation for \( y \): \[ y = 5 + t(3 - 5) = 2 \] Solving for \( t \): \[ 2 = 5 - 2t \quad \Rightarrow \quad t = \frac{3}{2} \]
Step 3: Use the value of \( t \) to find \( z \).
Substitute \( t = \frac{3}{2} \) into the equation for \( z \): \[ z = -1 + \frac{3}{2}(-2 + 2) = \frac{15}{2} \]
Step 4: Conclusion.
Thus, the \( z \)-coordinate of the point is \( \frac{15}{2} \).