Question

P is a point on the line joining the points \( (3,5,-1) \) and \( (6,3,-2) \). If \( y \)-coordinate of point P is 2, then \( x \)-coordinate will be

• A. \( -5 \)
• B. \( \frac{3}{2} \)
• C. \( \frac{15}{2} \)
• D. \( \frac{9}{2} \)

Hint:

For points on a line joining two points, use parametric form or section formula to easily find coordinates.

Answer 1

The Correct Option is C

Step 1: Write the given points.
The two points are:
\[ A(3,5,-1), \quad B(6,3,-2). \]

Step 2: Use parametric form of line.
A general point \( P \) on the line joining A and B can be written as:
\[ P = (3 + t(6-3),\ 5 + t(3-5),\ -1 + t(-2+1)). \]
\[ = (3+3t,\ 5-2t,\ -1-t). \]

Step 3: Use the given condition.
Given that the \( y \)-coordinate of point \( P \) is 2, so:
\[ 5 - 2t = 2. \]

Step 4: Solve for \( t \).
\[ 5 - 2t = 2 \Rightarrow 2t = 3 \Rightarrow t = \frac{3}{2}. \]

Step 5: Find the \( x \)-coordinate.
Substitute \( t = \frac{3}{2} \) into \( x = 3 + 3t \):
\[ x = 3 + 3\left(\frac{3}{2}\right). \]
\[ x = 3 + \frac{9}{2}. \]

Step 6: Simplify the expression.
\[ x = \frac{6}{2} + \frac{9}{2} = \frac{15}{2}. \]

Step 7: Final conclusion.
Thus, the required \( x \)-coordinate is \( \frac{15}{2} \).
Final Answer:
\[ \boxed{\frac{15}{2}}. \]