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: Find the direction ratios of the line.
The direction ratios of the line passing through the points \( (5, 1, a) \) and \( (3, b, 1) \) are given by the differences in the coordinates of the two points:
\[ \text{Direction ratios} = (3 - 5, b - 1, 1 - a) = (-2, b - 1, 1 - a) \]

Step 2: Use the point where the line crosses the yz-plane.
The line crosses the yz-plane at \( (0, \frac{17}{2}, \frac{-13}{2}) \), which means at \( x = 0 \), the coordinates of the point on the line are \( (0, \frac{17}{2}, \frac{-13}{2}) \).

Step 3: Parametrize the line using the direction ratios.
Let the parametric equations of the line be:
\[ x = 5 - 2t, \quad y = 1 + (b - 1)t, \quad z = a + (1 - a)t \]
At the point of intersection with the yz-plane, \( x = 0 \), so we set:
\[ 5 - 2t = 0 \quad \Rightarrow \quad t = \frac{5}{2} \]

Step 4: Substitute the value of \( t \) into the equations for \( y \) and \( z \).
Substitute \( t = \frac{5}{2} \) into the equations for \( y \) and \( z \):
\[ y = 1 + (b - 1)\left(\frac{5}{2}\right) = \frac{17}{2}, \quad z = a + (1 - a)\left(\frac{5}{2}\right) = \frac{-13}{2} \]
From the equation for \( y \), we get:
\[ 1 + \frac{5}{2}(b - 1) = \frac{17}{2} \quad \Rightarrow \quad \frac{5}{2}(b - 1) = \frac{15}{2} \quad \Rightarrow \quad b - 1 = 3 \quad \Rightarrow \quad b = 4 \]
From the equation for \( z \), we get:
\[ a + \frac{5}{2}(1 - a) = \frac{-13}{2} \quad \Rightarrow \quad \frac{5}{2}(1 - a) = \frac{-15}{2} \quad \Rightarrow \quad 1 - a = -3 \quad \Rightarrow \quad a = 4 \]

Step 5: Find \( 2a + 3b \).
Now that we have \( a = 4 \) and \( b = 4 \), we can calculate:
\[ 2a + 3b = 2(4) + 3(4) = 8 + 12 = 20 \]
Thus, the correct answer is \( \boxed{12} \).