Question

If the direction cosines of the line \(\frac{2x+5}{3}=\frac{2-3z}{-1};3y=-2\) are \(\frac{9}{\sqrt{85}},m,\frac{2}{\sqrt{85}}\) then the value of \(m\) is:

• A. \(\frac{9}{\sqrt{85}}\)
• B. \(0\)
• C. \(\frac{-2}{\sqrt{85}}\)
• D. \(1\)

Hint:

Convert the line to symmetric form and read off the direction ratios. A constant y means the y component is zero.

Answer 1

The Correct Option is B

Step 1: Observation
Since $y$ is fixed, the line lies in a plane parallel to the XZ-plane, so it has no y component.

Step 2: Check with Given Cosines
The given cosines $\dfrac9{\sqrt{85}}$ and $\dfrac2{\sqrt{85}}$ satisfy $\dfrac{81+4}{85}=1$.

Step 3: Conclude
Then $l^{2}+m^{2}+n^{2}=1$ forces $m=0$.

Final Answer:
The squares of the given cosines already add to 1, so m must be 0. \[ \boxed{\text{(B) }0} \]