Find the foot of the perpendicular from the point (0, 2, 3) on the line (-x - 3)/(-5) = (1 - y)/(-2) = (3z + 12)/(9) and hence find the length of the perpendicular.

MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2026: question 35(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section D, question 35(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(a) · 5 marks · Long answerThree Dimensional Geometry

Find the foot of the perpendicular from the point (0, 2, 3) on the line $\frac{- x - 3}{- 5} = \frac{1 - y}{- 2} = \frac{3 z + 12}{9}$ and hence find the length of the perpendicular.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

First rewrite the line so that each numerator is of the form $x - x_{1}$ , $y - y_{1}$ , $z - z_{1}$ with positive coefficients of x, y, z.

Hint 2

Take a general point Q on the line in terms of λ and write the direction ratios of PQ.

Hint 3

The dot product of PQ's direction ratios with the line's direction ratios must be zero; solve for λ.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Write the line in standard form: $\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3} = λ$ .
1 mark
Any point on the line is $Q (5 λ - 3, 2 λ + 1, 3 λ - 4)$ . The direction ratios of the line are $⟨ 5, 2, 3 ⟩$ .
½ mark
With $P (0, 2, 3)$ , the direction ratios of $P Q$ are $⟨ 5 λ - 3, 2 λ - 1, 3 λ - 7 ⟩$ .
½ mark
$P Q ⊥$ the line, so $5 (5 λ - 3) + 2 (2 λ - 1) + 3 (3 λ - 7) = 0$ .
This gives $38 λ - 38 = 0$ , so $λ = 1$ .
1 mark
Foot of the perpendicular: $Q = (5 - 3, 2 + 1, 3 - 4) = (2, 3, - 1)$ .
1 mark
Length $P Q = (2 - 0)^{2} + (3 - 2)^{2} + (- 1 - 3)^{2} = 4 + 1 + 16 = 21$ units.
1 mark

Answer: Foot of perpendicular $(2, 3, - 1)$ ; length $21$ units

Where the marks are

Converts the line to standard form $\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}$ (1 mark)
Writes the general point $(5 λ - 3, 2 λ + 1, 3 λ - 4)$ (½ mark)
Writes the direction ratios of $P Q$ (½ mark)
Uses the perpendicularity condition to get $λ = 1$ (1 mark)
Finds the foot $(2, 3, - 1)$ (1 mark)
Finds the length $21$ (1 mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)
Topic: Foot of Perpendicular and Image of a Point

Practise this chapterBoard-style questions from Three Dimensional Geometry, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 35(a)

Hints

Step-by-step solution

Revise the chapter

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