Verify that lines given by r=(1-λ)i+(λ-2)j+(3-2λ)k and r=(μ+1)i+(2μ-1)j-(2μ+1)k are skew lines. Hence, find shortest distance between the lines.

The lines are skew; shortest distance = (8)/(√(29)) units

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2025: question 29(a)

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, 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

Verify that lines given by $r = (1 - λ) \hat{i} + (λ - 2) \hat{j} + (3 - 2 λ) \hat{k}$ and $r = (μ + 1) \hat{i} + (2 μ - 1) \hat{j} - (2 μ + 1) \hat{k}$ are skew lines. Hence, find shortest distance between the lines.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

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

Hint 1

Separate each line into a fixed point plus a parameter times a direction vector.

Hint 2

Skew lines are neither parallel nor intersecting. A non-zero value of $(a_{2} - a_{1}) \cdot (b_{1} \times b_{2})$ shows they do not meet.

Hint 3

Divide the absolute value of that triple product by $∣ b_{1} \times b_{2} ∣$ .

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.

Rewrite the lines in vector form: $r = (\hat{i} - 2 \hat{j} + 3 \hat{k}) + λ (- \hat{i} + \hat{j} - 2 \hat{k})$ and $r = (\hat{i} - \hat{j} - \hat{k}) + μ (\hat{i} + 2 \hat{j} - 2 \hat{k})$ . So $a_{1} = \hat{i} - 2 \hat{j} + 3 \hat{k}$ , $b_{1} = - \hat{i} + \hat{j} - 2 \hat{k}$ , $a_{2} = \hat{i} - \hat{j} - \hat{k}$ , $b_{2} = \hat{i} + 2 \hat{j} - 2 \hat{k}$ . The direction ratios $- 1, 1, - 2$ and $1, 2, - 2$ are not proportional, so the lines are not parallel.
½ mark
$a_{2} - a_{1} = \hat{j} - 4 \hat{k}$ .
½ mark
$b_{1} \times b_{2} = \hat{i} - 1 1 \hat{j} 12 \hat{k} - 2 - 2 = \hat{i} (- 2 + 4) - \hat{j} (2 + 2) + \hat{k} (- 2 - 1) = 2 \hat{i} - 4 \hat{j} - 3 \hat{k}$ .
½ mark
Compute $(a_{2} - a_{1}) \cdot (b_{1} \times b_{2}) = 0 - 4 + 12 = 8 \neq = 0$ . So the lines do not intersect. Being neither parallel nor intersecting, the lines are skew.
½ mark
Shortest distance $= \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣} = \frac{8}{4 + 16 + 9} = \frac{8}{29}$ units.
1 mark

Answer: The lines are skew; shortest distance $= \frac{8}{29}$ units

Where the marks are

Rewrites the lines as $a + λ b$ and notes they are not parallel (½ mark)
Finds $a_{2} - a_{1}$ (½ mark)
Finds $b_{1} \times b_{2} = 2 \hat{i} - 4 \hat{j} - 3 \hat{k}$ (½ mark)
Shows the scalar triple product is $8 \neq = 0$ , so the lines are skew (½ mark)
Gets shortest distance $\frac{8}{29}$ (1 mark)

Revise the chapter

Chapter: Three Dimensional Geometry (notes, formulas, where marks are lost and every past question)
Topic: Shortest Distance Between Two Lines

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 paper2025 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2025: question 29(a)

Hints

Step-by-step solution

Revise the chapter

More questions from Three Dimensional Geometry