Find the point Q on the line (2x+4)/(6) = (y+1)/(2) = (-2z+6)/(-4) at a distance of 3√(2) from the point P(1, 2, 3).

Q(-2,-1,3) or Q (56/17,43/17,111/17 )

MathsCBSE Class 12 · Board exam 2025, set 65/7/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2025: question 35(a)

Set in the CBSE Class 12 board exam of 2025, paper 65/7/1, 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 point Q on the line $\frac{2 x + 4}{6} = \frac{y + 1}{2} = \frac{- 2 z + 6}{- 4}$ at a distance of $32$ from the point P(1, 2, 3).

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/7/1.

Hints

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

Hint 1

First make the coefficients of $x$ , $y$ , $z$ equal to 1 in each numerator so the line is in standard form.

Hint 2

Write a general point on the line in terms of a parameter $λ$ .

Hint 3

Use the distance formula with $P Q^{2} = 18$ and solve the resulting quadratic in $λ$ .

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: $\frac{x + 2}{3} = \frac{y + 1}{2} = \frac{z - 3}{2} = λ$ . Any point on it is $Q (3 λ - 2, 2 λ - 1, 2 λ + 3)$ for some $λ \in R$ .
2 marks
Since $P Q = 32$ , we have $P Q^{2} = 18$ :
$(3 λ - 3)^{2} + (2 λ - 3)^{2} + (2 λ)^{2} = 18$ .
1 mark
Expanding: $9 λ^{2} - 18 λ + 9 + 4 λ^{2} - 12 λ + 9 + 4 λ^{2} = 18 \Rightarrow 17 λ^{2} - 30 λ = 0 \Rightarrow λ = 0$ or $λ = \frac{30}{17}$ .
1 mark
For $λ = 0$ : $Q (- 2, - 1, 3)$ . For $λ = \frac{30}{17}$ : $Q (\frac{90}{17} - 2, \frac{60}{17} - 1, \frac{60}{17} + 3) = Q (\frac{56}{17}, \frac{43}{17}, \frac{111}{17})$ .
1 mark

Answer: $Q (- 2, - 1, 3)$ or $Q (\frac{56}{17}, \frac{43}{17}, \frac{111}{17})$

Where the marks are

Converts the line to standard form and writes the general point $(3 λ - 2, 2 λ - 1, 2 λ + 3)$ (2 marks)
Sets up $P Q^{2} = 18$ correctly (1 mark)
Solves $17 λ^{2} - 30 λ = 0$ to get $λ = 0$ or $\frac{30}{17}$ (1 mark)
States both points Q (1 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 paper2025 board paper, set 65/7/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 35(a)

Hints

Step-by-step solution

Revise the chapter

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