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COMEDK 2025 Evening Solved Paper

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Question : 164 of 180

Marks: +1, -0

The image of a point

P(3,5,3)

\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}

P^{\prime}(a, b, c)

a+b+c=

$-$ 17
$-$ 7
3
7

Solution:

Concept:

Reflection of a point across a line uses the foot of the perpendicular, then

P' = 2H - P

.

Explanation:

The given line is

L: \frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}

.

A point on

L

is

A = (0,1,2)

and its direction vector is

\mathbf{v} = (1,2,3)

.

Point

P = (3,5,3)

; compute

\overrightarrow{AP} = P - A = (3,4,1)

.

Project

\overrightarrow{AP}

onto

\mathbf{v}

to get

t = \frac{\overrightarrow{AP} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} = \frac{3\cdot1+4\cdot2+1\cdot3}{1^2+2^2+3^2} = \frac{14}{14} = 1

.

The foot of the perpendicular is

H = A + t\mathbf{v} = (0,1,2) + (1,2,3) = (1,3,5)

.

The reflection

P'

is

P' = 2H - P = (2\cdot1-3,\;2\cdot3-5,\;2\cdot5-3) = (-1,1,7)

.

Hence

a = -1

,

b = 1

,

c = 7

, and

a+b+c = -1+1+7 = 7

.

Answer:

7

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