The length of perpendicular drawn from the point \((1, 2, 3)\) on line \(\dfrac{x}{0} = \dfrac{y}{1} = \dfrac{z}{0}\) is
The length of perpendicular drawn from the point \((1, 2, 3)\) on line \(\dfrac{x}{0} = \dfrac{y}{1} = \dfrac{z}{0}\) is
Show Hint
- \(2\)
- \(6\)
- \(\sqrt{10}\)
- \(\sqrt{14}\)
The Correct Option is C
Solution and Explanation
The length of the perpendicular from a point to a line is the shortest distance from the point to the line.
We find the foot of the perpendicular on the line and then use the distance formula.
Step 2: Read the line:
The line \(\dfrac{x}{0} = \dfrac{y}{1} = \dfrac{z}{0}\) passes through the origin with direction ratios \((0, 1, 0)\).
So it is the y-axis. Its points are \((0, \lambda, 0)\).
Step 3: Find the foot of the perpendicular:
Let the foot be \(M(0, \lambda, 0)\) and the given point be \(P(1, 2, 3)\). Then:
\[ \overrightarrow{PM} = (-1, \ \lambda - 2, \ -3) \]
\(PM\) is perpendicular to the direction \((0, 1, 0)\), so their dot product is zero:
\[ (-1)(0) + (\lambda - 2)(1) + (-3)(0) = 0 \Rightarrow \lambda = 2 \]
So the foot is \(M(0, 2, 0)\).
Step 4: Find the length:
\[ PM = \sqrt{(1-0)^2 + (2-2)^2 + (3-0)^2} = \sqrt{1 + 0 + 9} = \sqrt{10} \]
Step 5: Why the other options are wrong:
Option (A) 2 is only the y-coordinate of \(P\), which is the distance along the line, not the distance to the line.
Option (B) 6 and option (D) \(\sqrt{14}\) come from adding wrong terms. \(\sqrt{14}\) is the distance of \(P\) from the origin, \(\sqrt{1+4+9}\).
Final Answer:
The perpendicular distance is \(\sqrt{10}\), option (C). \[ \boxed{\sqrt{10}} \]
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