Find the point of intersection of the line r=(3 i+ k)+μ( i+ j+ k) and the line through (2,-1,1) parallel to the z-axis. How far is this point from 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(B)

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

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

Find the point of intersection of the line $r = (3 \hat{i} + \hat{k}) + μ (\hat{i} + \hat{j} + \hat{k})$ and the line through $(2, - 1, 1)$ parallel to the z-axis. How far is this point from 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

Write a general point on each line using parameters.

Hint 2

Distance of $(x, y, z)$ from the z-axis is $x^{2} + y^{2}$ .

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 $(2, - 1, 1)$ parallel to the z-axis: $r = (2 \hat{i} - \hat{j} + \hat{k}) + λ \hat{k}$ ; any point $P (2, - 1, 1 + λ)$ .
1 mark
Any point on the given line: $A (3 + μ, μ, 1 + μ)$ . Equating: $3 + μ = 2$ and $μ = - 1$ $\Rightarrow μ = - 1$ (consistent).
1 mark
Point of intersection: $(3 - 1, - 1, 1 - 1) = (2, - 1, 0)$ .
½ mark
Distance of $(2, - 1, 0)$ from the z-axis $= x^{2} + y^{2} = 4 + 1 = 5$ units.
½ mark

Answer: Point $(2, - 1, 0)$ ; distance from z-axis $5$ units

Where the marks are

Equation/general point of line parallel to z-axis (1 mark)
Equates general points and gets $μ = - 1$ (1 mark)
Point of intersection $(2, - 1, 0)$ (½ mark)
Distance $5$ units (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(B)

Hints

Step-by-step solution

Revise the chapter

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