Question

If the points \((1,1,μ)\) and \((-3,0,1)\) are equidistant from the plane \(\overset{⃗}{r}\cdot (3\hat{i}+4\hat{j}-12\hat{k})+13 = 0\), then the value of \(μ\) are

• A. \(1,\frac{-7}{3}\)
• B. \(1,\frac{7}{3}\)
• C. \(-1,\frac{7}{3}\)
• D. \(1,\frac{3}{7}\)

Hint:

Set the two distances from the plane equal.

Answer 1

The Correct Option is B

Step 1: Plane in Cartesian form
\(3x+4y-12z+13=0\) and \(\sqrt{9+16+144}=13\).

Step 2: Distance of \((-3,0,1)\)
\(\frac{|-9+0-12+13|}{13}=\frac{8}{13}\).

Step 3: Distance of \((1,1,\mu)\)
\(\frac{|3+4-12\mu+13|}{13}=\frac{|20-12\mu|}{13}\).

Step 4: Equate
\(|20-12\mu|=8\) gives \(12\mu=12\) or \(12\mu=28\), so \(\mu=1\) or \(\frac73\). Option (B).

Final Answer:
\(\mu=1,\frac73\), option (B). \[ \boxed{\text{(B)}} \]