Question

The line passing through the point \( (5, 1, a) \) and \( (3, b, 1) \) crosses the yz-plane at \( (0, \frac{17}{2}, \frac{-13}{2}) \), then the value of \( 2a + 3b \) is:

• A. 10
• B. 12
• C. 22
• D. 24

Hint:

When finding the value of a parameter for a line, use the parametric equations and the given point where the line crosses a coordinate plane to solve for the unknown variables.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The three points given must be collinear because the line passing through the first two also passes through the third (where it crosses the plane).
Step 2: Key Formula or Approach:
For points \( P_1(x_1, y_1, z_1) \), \( P_2(x_2, y_2, z_2) \), and \( P_3(x_3, y_3, z_3) \) to be collinear:
\[ \frac{x_1 - x_2}{x_2 - x_3} = \frac{y_1 - y_2}{y_2 - y_3} = \frac{z_1 - z_2}{z_2 - z_3} \]
Step 3: Detailed Explanation:
Let \( P_1 = (5, 1, a) \), \( P_2 = (3, b, 1) \), and \( P_3 = (0, 17/2, -13/2) \).
\[ \frac{5 - 3}{3 - 0} = \frac{1 - b}{b - 17/2} = \frac{a - 1}{1 - (-13/2)} \] \[ \frac{2}{3} = \frac{1 - b}{b - 17/2} = \frac{a - 1}{15/2} \] Solving for \( b \):
\( 2(b - 17/2) = 3(1 - b) \)
\( 2b - 17 = 3 - 3b \Rightarrow 5b = 20 \Rightarrow b = 4 \).
Solving for \( a \):
\( \frac{2}{3} = \frac{a - 1}{15/2} \Rightarrow \frac{2}{3} = \frac{2(a - 1)}{15} \)
\( 10 = 2(a - 1) \Rightarrow 5 = a - 1 \Rightarrow a = 6 \).
Calculation of \( 2a + 3b \):
\( 2(6) + 3(4) = 12 + 12 = 24 \).
Step 4: Final Answer:
The value is 24.