Question 35 (Choice 1) - CBSE Class 12 Sample Paper for 2024 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at April 16, 2024 by Teachoo

Find the coordinates of the image of the point (1 , 6, 3) with respect to the line r ⃗=(j ˆ+2k ˆ)+λ(i ˆ+2j ˆ+3k ˆ); where ' λ ' is a scalar. Also, find the distance of the image from the y-axis.

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Let Point P be (1, 6, 3)
Let Q (a, b, c) be the image of
point P (1, 6, 3) in the line 𝒓 ⃗
Since line is a mirror
Point P & Q are at equal distance from line AB, i.e. PR = QR, i.e. R is the mid point of PQ
Image is formed perpendicular to mirror i.e. line PQ is perpendicular to line 𝒓 ⃗
Given line is 𝑟 ⃗=(𝑗 ˆ+2𝑘 ˆ)+𝜆(𝑖 ˆ+2𝑗 ˆ+3𝑘 ˆ)
In cartesian form
(𝒙 − 𝟎)/𝟏 = (𝒚 − 𝟏)/𝟐 = (𝒛 − 𝟐)/𝟑
Since PQ ⊥ Line 1 (𝑙_1)
∴ PR ⊥ Line 1 (𝒍_𝟏)
Coordinates of R
Since R lies of line 𝑙_1
∴ (𝑥 − 0)/1 = (𝑦 − 1)/2 = (𝑧 − 2)/3 = 𝜆
∴ x = 𝝀 , y = 2𝝀 + 1 and z = 3𝝀 + 2Direction ratios of Line 𝒍_𝟏
Since equation of lines is
(𝒙 − 𝟎)/𝟏 = (𝒚 − 𝟏)/𝟐 = (𝒛 − 𝟐)/𝟑
Direction ratios are 1, 2, 3
Direction ratios of Line PR
Coordinates of P (1, 6, 3)
Coordinates of R R (𝝀, 2𝝀 + 1, 3𝝀 + 2)
Direction ratios are
𝜆 – 1, 2𝜆 + 1 – 6 & 3𝜆 + 2 – 3
i.e. 𝝀 – 1, 2𝝀 – 5 & 3𝝀 – 1
14𝜆 – 14 = 0
14𝜆 = 14
𝜆 = 14/14
𝝀 = 1
Now,
Coordinates of R = (𝜆, 2𝜆 + 1, 3𝜆 + 2)
= (1, 2(1) + 1, 3(1) + 2)
= (𝟏, 3, 5)
Since R is the midpoint of PQ
Coordinates of R = ((𝟏 + 𝒂)/𝟐 " , " (𝟔 + 𝒃)/𝟐 " , " (𝟑 + 𝒄)/𝟐)
(1, 3, 5)= ((𝟏 + 𝒂)/𝟐 " , " (𝟔 + 𝒃)/𝟐 " , " (𝟑 + 𝒄)/𝟐)
1 = (1+𝑎)/2 , 3 = (6+𝑏)/2 , 5 = (3+𝑐)/2
2 = 1 + a, 6 = 6 + b , 10 = 3 + c
∴ a = 1, b = 0 and c = 7
Hence, Q(1,0,7) is the required image of P
Finding the distance of the image from the 𝒚-axis.
Distance of Q(1, 0, 7) from the 𝑦-axis
= Distance of parallel point Y and point Q
= Distance of point Y (0, 0, 0) and point Q (1, 0, 7)
=√(〖(0−1)〗^2 + 〖(0−0)〗^2 + 〖(0−7)〗^2 )
=√(1+49)
=√𝟓𝟎 units
asdf

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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