Question:
The length of perpendicular drawn from the point \( (3, 4, 2) \) on the line \( \frac{x}{0} = \frac{y}{0} = \frac{z}{1} \) is
The length of perpendicular drawn from the point \( (3, 4, 2) \) on the line \( \frac{x}{0} = \frac{y}{0} = \frac{z}{1} \) is
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Tip 1: Distance from \( z \)-axis is \( \sqrt{x^2 + y^2} \).
Tip 2: Distance from \( y \)-axis is \( \sqrt{x^2 + z^2} \).
Tip 3:Distance from \( x \)-axis is \( \sqrt{y^2 + z^2} \).
Tip 2: Distance from \( y \)-axis is \( \sqrt{x^2 + z^2} \).
Tip 3:Distance from \( x \)-axis is \( \sqrt{y^2 + z^2} \).
Updated On: Sep 10, 2026
- \( 2 \)
- \( 9 \)
- \( 5 \)
- \( \sqrt{29} \)
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The Correct Option is C
Solution and Explanation
Concept:
• The equation of the line is \( \frac{x}{0} = \frac{y}{0} = \frac{z}{1} \).
• This line passes through the origin \( (0, 0, 0) \) and has direction ratios \( (0, 0, 1) \), which identifies it as the \( z \)-axis.
• The perpendicular distance from a point \( (x, y, z) \) to the \( z \)-axis is \( \sqrt{x^2 + y^2} \).
Step 1: Identify the target line
The denominators of the symmetric form are \( 0, 0, 1 \). This indicates the line is parallel to the vector \( \hat{k} \). Since it passes through \( (0, 0, 0) \), the line is exactly the \( z \)-axis.
Step 2: Determine the foot of the perpendicular
Let the given point be \( P(3, 4, 2) \). The foot of the perpendicular \( M \) from \( P \) onto the \( z \)-axis will have coordinates \( (0, 0, z_P) \). Therefore, \( M = (0, 0, 2) \).
Step 3: Calculate the distance between the point and the foot
Using the distance formula: \[ \text{Length} = \sqrt{(3 - 0)^2 + (4 - 0)^2 + (2 - 2)^2} \] \[ \text{Length} = \sqrt{3^2 + 4^2 + 0^2} \]
Step 4: Final computation
\[ \text{Length} = \sqrt{9 + 16} \] \[ \text{Length} = \sqrt{25} = 5 \]
• The equation of the line is \( \frac{x}{0} = \frac{y}{0} = \frac{z}{1} \).
• This line passes through the origin \( (0, 0, 0) \) and has direction ratios \( (0, 0, 1) \), which identifies it as the \( z \)-axis.
• The perpendicular distance from a point \( (x, y, z) \) to the \( z \)-axis is \( \sqrt{x^2 + y^2} \).
Step 1: Identify the target line
The denominators of the symmetric form are \( 0, 0, 1 \). This indicates the line is parallel to the vector \( \hat{k} \). Since it passes through \( (0, 0, 0) \), the line is exactly the \( z \)-axis.
Step 2: Determine the foot of the perpendicular
Let the given point be \( P(3, 4, 2) \). The foot of the perpendicular \( M \) from \( P \) onto the \( z \)-axis will have coordinates \( (0, 0, z_P) \). Therefore, \( M = (0, 0, 2) \).
Step 3: Calculate the distance between the point and the foot
Using the distance formula: \[ \text{Length} = \sqrt{(3 - 0)^2 + (4 - 0)^2 + (2 - 2)^2} \] \[ \text{Length} = \sqrt{3^2 + 4^2 + 0^2} \]
Step 4: Final computation
\[ \text{Length} = \sqrt{9 + 16} \] \[ \text{Length} = \sqrt{25} = 5 \]
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