Question

The plane passing through the point \((5,1,2)\) and perpendicular to the line \(2x-4=y-4=z-5\) will meet the line at the point

• A. \((3,2,2)\)
• B. \((3,2,1)\)
• C. \((2,3,1)\)
• D. \((1,2,3)\)

Hint:

Take the direction ratios of the line as the normal of the plane and find where the line meets it.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The plane is perpendicular to the line, so the direction ratios of the line give the normal of the plane.

Step 2: Direction ratios and plane
\(2x-4=y-4=z-5\) gives \(\frac{x-2}{1/2}=\frac{y-4}{1}=\frac{z-5}{1}\), i.e. direction ratios \((1,2,2)\).
Plane through \((5,1,2)\): \((x-5)+2(y-1)+2(z-2)=0\Rightarrow x+2y+2z=11\).

Step 3: Find the meeting point
Points on the line: \((2+\lambda,\ 4+2\lambda,\ 5+2\lambda)\).
\(2+\lambda+8+4\lambda+10+4\lambda=11\Rightarrow9\lambda=-9\Rightarrow\lambda=-1\).

Step 4: Coordinates
The point is \((1,2,3)\).

Final Answer:
The plane meets the line at the point (1, 2, 3). \[ \boxed{\text{(D) }(1,2,3)} \]