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CBSE Board Examination 2026 · Section A · 1 mark

Q16: Perpendicular from (2,5,7)(2, 5, 7) to the line x1=y0=z0\frac{x}{1} = \frac{y}{0} = \frac{z}{0}

Board 2026Ch 11 · Three Dimensional Geometry
The question

The length of perpendicular drawn from point (2,5,7)(2, 5, 7) on line x1=y0=z0\frac{x}{1} = \frac{y}{0} = \frac{z}{0} is

(A) 22

(B) 55

(C) 74\sqrt{74}

(D) 78\sqrt{78}

Exam answer

The line passes through the origin with direction ratios 1, 0, 0, so it is the xx-axis.

Why?

The line is in the form

x−01=y−00=z−00\frac{x - 0}{1} = \frac{y - 0}{0} = \frac{z - 0}{0}

so it passes through (0,0,0)(0, 0, 0) with direction ratios 1, 0, 0. A zero below means that coordinate never changes: y=0y = 0 and z=0z = 0 all along the line.

The foot of the perpendicular from (2,5,7)(2, 5, 7) is (2,0,0)(2, 0, 0)⁠.

Why?

A general point of the line is Q(λ,0,0)Q(\lambda, 0, 0)⁠. From P(2,5,7)P(2, 5, 7)⁠, PQ→\overrightarrow{PQ} has direction ratios λ−2,−5,−7\lambda - 2, -5, -7⁠. It is perpendicular to the line when

(λ−2)(1)+0+0=0(\lambda - 2)(1) + 0 + 0 = 0

so λ=2\lambda = 2⁠, and the foot is (2,0,0)(2, 0, 0)⁠.

Length=02+52+72=25+49=74\begin{aligned} \text{Length} &= \sqrt{0^2 + 5^2 + 7^2} \\ &= \sqrt{25 + 49} = \sqrt{74} \end{aligned}
Why?

The distance formula between (2,5,7)(2, 5, 7) and (2,0,0)(2, 0, 0)⁠: the differences are 0,5,70, 5, 7⁠.

(2−2)2+(5−0)2+(7−0)2=02+52+72=0+25+49=74\begin{aligned} &\sqrt{(2 - 2)^2 + (5 - 0)^2 + (7 - 0)^2} \\ &= \sqrt{0^2 + 5^2 + 7^2} \\ &= \sqrt{0 + 25 + 49} = \sqrt{74} \end{aligned}

Watch out: 78=4+25+49\sqrt{78} = \sqrt{4 + 25 + 49}⁠, option (D), is the distance from the origin, not from the line. And 2, option (A), is the foot's xx-coordinate.

Answer: (C) 74\sqrt{74}

Where this comes from

Chapter 11, Three Dimensional Geometry: Perpendicular distance of a point from a line.

Question type: the foot of a perpendicular

How to spot it: click to see

Type 10. Beyond the book, but in CBSE papers: the foot of a perpendicular

Spot it: "Find the foot of the perpendicular from a point to a line", "the image of a point in a line", or "do the lines meet?". This book has none of these, but recent CBSE sample papers ask all three.

Here: x1=y0=z0\frac{x}{1} = \frac{y}{0} = \frac{z}{0} is the x-axis, so the foot from (2,5,7)(2, 5, 7) is (2,0,0)(2, 0, 0)⁠, and the length is 52+72=74\sqrt{5^2 + 7^2} = \sqrt{74}⁠.

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