Question

If the lines \(\frac{x-5}{5m+2} = \frac{2-y}{5} = \frac{1-z}{-1}\) and \(x = \frac{2y+1}{4m} = \frac{1-z}{-3}\) are perpendicular to each other, then the value of m is ...

• A. \(1\)
• B. \(0\)
• C. \(2\)
• D. \(-1\)

Hint:

Get direction ratios (5m+2, -5, 1) and (1, 2m, 3).

Answer 1

The Correct Option is A

Step 1: Compare with options:
Test $m=1$: line 1 has direction $(7,-5,1)$ and line 2 has $(1,2,3)$.

Step 2: Dot product:
$7-10+3=0$. So they are perpendicular.

Step 3: Test another:
$m=0$: $(2,-5,1)\cdot(1,0,3)=5\ne0$. $m=2$: $(12,-5,1)\cdot(1,4,3)=-5\ne0$. $m=-1$: $(-3,-5,1)\cdot(1,-2,3)=10\ne0$. Only $m=1$ works.

Final Answer:
Only m = 1 gives a zero dot product. \[ \boxed{A} \]