Find the distance of the point (2,-1,3) from the line r=(2 i- j+2 k)+μ(3 i+6 j+2 k) measured parallel to the z-axis.

MathsCBSE Class 12 · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2025-26: question 29(A)

From the official CBSE sample question paper for 2025-26, section C, question 29(A), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(A) · 3 marks · Short answerThree Dimensional Geometry

Find the distance of the point $(2, - 1, 3)$ from the line $r = (2 \hat{i} - \hat{j} + 2 \hat{k}) + μ (3 \hat{i} + 6 \hat{j} + 2 \hat{k})$ measured parallel to the z-axis.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

'Measured parallel to the z-axis' means: draw a line through the point in the direction $\hat{k}$ .

Hint 2

Find where that line meets the given line, then find the distance between the two points.

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.

Line through $(2, - 1, 3)$ parallel to the z-axis: $r = (2 \hat{i} - \hat{j} + 3 \hat{k}) + λ \hat{k}$ .
1 mark
Any point on it: $P (2, - 1, 3 + λ)$ .
½ mark
Any point on the given line: $Q (2 + 3 μ, - 1 + 6 μ, 2 + 2 μ)$ .
½ mark
For intersection, $P = Q$ : $2 + 3 μ = 2 \Rightarrow μ = 0$ (also $- 1 + 6 μ = - 1$ holds). Point of intersection $(2, - 1, 2)$ .
½ mark
Required distance $= 0^{2} + 0^{2} + (3 - 2)^{2} = 1$ unit.
½ mark

Answer: 1 unit

Where the marks are

Equation of line through the point parallel to z-axis (1 mark)
General point on this line (½ mark)
General point on given line (½ mark)
Finds $μ = 0$ and point $(2, - 1, 2)$ (½ mark)
Distance 1 unit (½ mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)

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 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 29(A)

Hints

Step-by-step solution

Revise the chapter

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