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.

The rest of the post is locked. Join Teachoo Black to see the full post.

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

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, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!