Find the foot of the perpendicular drawn from the point (2,1,4) on the line x-2=(2y+3)/(4)=2-z. Also find the equation of the line passing through this point and the point (2,1,4).

Foot (52,-12,32 ); line (x-2)/(1)=(y-1)/(-3)=(z-4)/(-5)

MathsCBSE Class 12 · Sample paper, session 2026-27 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 30(A)

From the official CBSE sample question paper for 2026-27, section C, question 30(A), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30(A) · 3 marks · Short answerThree Dimensional Geometry

Find the foot of the perpendicular drawn from the point $(2, 1, 4)$ on the line $x - 2 = \frac{2 y + 3}{4} = 2 - z$ . Also find the equation of the line passing through this point and the point $(2, 1, 4)$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

Put the line in symmetric form equal to $k$ and write a general point; note $\frac{2 y + 3}{4} = \frac{y + 3/2}{2}$ and $2 - z = \frac{z - 2}{- 1}$ .

Hint 2

Make $A P$ perpendicular to the line using a zero dot product of direction ratios.

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.

Let $x - 2 = \frac{2 y + 3}{4} = 2 - z = k$ . A general point on the line is $P (k + 2, \frac{4 k - 3}{2}, 2 - k)$ . The line's direction ratios are $⟨ 1, 2, - 1 ⟩$ .
½ mark
Direction ratios of $A P$ , where $A (2, 1, 4)$ : $⟨ k, \frac{4 k - 5}{2}, - k - 2 ⟩$
½ mark
$A P ⊥$ the line: $1 (k) + 2 (\frac{4 k - 5}{2}) + (- 1) (- k - 2) = 0 \Rightarrow 6 k - 3 = 0 \Rightarrow k = \frac{1}{2}$
½ mark
Foot of the perpendicular: $P (\frac{5}{2}, - \frac{1}{2}, \frac{3}{2})$
½ mark
Direction ratios of $A P$ : $⟨ \frac{1}{2}, - \frac{3}{2}, - \frac{5}{2} ⟩$ , i.e. $⟨ 1, - 3, - 5 ⟩$
½ mark
Equation of the line through $(2, 1, 4)$ and the foot: $\frac{x - 2}{1} = \frac{y - 1}{- 3} = \frac{z - 4}{- 5}$
½ mark

Answer: Foot $(\frac{5}{2}, - \frac{1}{2}, \frac{3}{2})$ ; line $\frac{x - 2}{1} = \frac{y - 1}{- 3} = \frac{z - 4}{- 5}$

Where the marks are

Writes a general point on the line (½ mark)
Finds the direction ratios of $A P$ (½ mark)
Uses perpendicularity to get $k = \frac{1}{2}$ (½ mark)
Finds the foot $(\frac{5}{2}, - \frac{1}{2}, \frac{3}{2})$ (½ mark)
Gets the direction ratios $⟨ 1, - 3, - 5 ⟩$ (½ mark)
Writes the equation of the line (½ 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 paperSample paper 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 30(A)

Hints

Step-by-step solution

Revise the chapter

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