The image of point P(x, y, z) with respect to line x/1 = (y-1)/(2) = (z-2)/(3) is P'(1, 0, 7). Find the coordinates of point P.

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2024: question 35

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section D, question 35, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35 · 5 marks · Long answerThree Dimensional Geometry

The image of point $P (x, y, z)$ with respect to line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ is $P^{'} (1, 0, 7)$ . Find the coordinates of point P.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

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

Hint 1

The line is the perpendicular bisector of $P P^{'}$ , so first find the foot of the perpendicular from $P^{'}$ to the line.

Hint 2

Write a general point on the line in terms of $λ$ ; the vector from $P^{'}$ to it must be perpendicular to $(1, 2, 3)$ .

Hint 3

The foot of the perpendicular is the mid-point of $P$ and $P^{'}$ .

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.

Any point on the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} = λ$ is $(λ, 2 λ + 1, 3 λ + 2)$ . Let $M (λ, 2 λ + 1, 3 λ + 2)$ be the foot of the perpendicular from $P^{'}$ (and hence from $P$ ) to the line.
1 mark
Direction ratios of $P^{'} M$ are $(λ - 1, 2 λ + 1, 3 λ - 5)$ .
1 mark
$P^{'} M ⊥$ line, whose direction ratios are $1, 2, 3$ :
$1 (λ - 1) + 2 (2 λ + 1) + 3 (3 λ - 5) = 0 \Rightarrow 14 λ - 14 = 0 \Rightarrow λ = 1$
1 mark
So the foot of the perpendicular is $M (1, 3, 5)$ .
1 mark
$M$ is the mid-point of $P P^{'}$ [the line is the perpendicular bisector of $P P^{'}$ ]:
$\frac{x + 1}{2} = 1, \frac{y + 0}{2} = 3, \frac{z + 7}{2} = 5 \Rightarrow x = 1, y = 6, z = 3$
Hence $P (1, 6, 3)$ .
1 mark

Answer: $P (1, 6, 3)$

Where the marks are

Takes a general point $M (λ, 2 λ + 1, 3 λ + 2)$ on the line as the foot of the perpendicular (1 mark)
Writes the direction ratios of $P^{'} M$ (1 mark)
Uses perpendicularity to get $λ = 1$ (1 mark)
Finds the foot $M (1, 3, 5)$ (1 mark)
Uses the mid-point formula to get $P (1, 6, 3)$ (1 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 paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 35

Hints

Step-by-step solution

Revise the chapter

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