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:

Write the line in parametric form from its two points and use the point where x = 0 to get a and b.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The yz-plane is x = 0. The line through \((5,1,a)\) and \((3,b,1)\) has direction \((-2,\,b-1,\,1-a)\).

Step 2: Parametric form
\(x=5-2t,\ y=1+t(b-1),\ z=a+t(1-a)\)
Setting \(x=0\) gives \(t=\dfrac52\).

Step 3: Find b and a
\(y=1+\dfrac52(b-1)=\dfrac{17}{2}\Rightarrow b-1=3\Rightarrow b=4\)
\(z=a+\dfrac52(1-a)=-\dfrac{13}{2}\Rightarrow-\dfrac32a=-9\Rightarrow a=6\)

Step 4: Compare the options
\(2a+3b=12+12=24\), the fourth option.

Final Answer:
The value of 2a + 3b is 24. \[ \boxed{\text{(D) }24} \]