Question 10 (Choice 2) - CBSE Class 12 Sample Paper for 2022 Boards (For Term 2) - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards
Last updated at Jan. 23, 2022 by Teachoo
Find the vector and the Cartesian equations of the plane containing the point ๐ย ฬ+2๐ย ฬโ๐ย ฬ and parallel to the lines ๐ย โ = (๐ย ฬ+2๐ย ฬ+2๐ย ฬ ) + s (2๐ย ฬโ3๐ย ฬ+2๐ย ฬ ) = 0 and ๐ย โ = (3๐ย ฬ+๐ย ฬโ2๐ย ฬ ) + t (๐ย ฬโ3๐ย ฬ+๐ย ฬ ) = 0
Question 10 (Choice 2) Find the vector and the Cartesian equations of the plane containing the point ๐ ฬ+2๐ ฬโ๐ ฬ and parallel to the lines ๐ โ = (๐ ฬ+2๐ ฬ+2๐ ฬ ) + s (2๐ ฬโ3๐ ฬ+2๐ ฬ ) = 0 and ๐ โ = (3๐ ฬ+๐ ฬโ2๐ ฬ ) + t (๐ ฬโ3๐ ฬ+๐ ฬ ) = 0 Given that
Plane is parallel to lines
๐ โ = (๐ ฬ+2๐ ฬ+2๐ ฬ ) + s (2๐ ฬโ3๐ ฬ+2๐ ฬ )
and ๐ โ = (3๐ ฬ+๐ ฬโ2๐ ฬ ) + t (๐ ฬโ3๐ ฬ+๐ ฬ )
Thus,
Normal will be perpendicular to both their parallel vectors
โด Normal of plane = (๐๐ ฬโ๐๐ ฬ+๐๐ ฬ ) ร (๐ ฬโ๐๐ ฬ+๐ ฬ )
= |โ 8(๐ ฬ&๐ ฬ&๐ ฬ@2&โ3&2@1&โ3&1)|
= ๐ ฬ (โ3(1) โ (โ3)(2)) โ ๐ ฬ (2(1) โ 1(2)) + ๐ ฬ(2(โ3) โ 1(โ3))
= ๐ ฬ (โ3 + 6) โ ๐ ฬ (2 โ 2) + ๐ ฬ(โ6 + 3)
= 3๐ ฬ โ 3๐ ฬ
Now,
Equation of plane passing through point A whose position vector is ๐ โ & perpendicular to ๐ โ is
(๐ โ โ ๐ โ) . ๐ โ = 0
Thus,
Equation of plane passing through ๐ โ =๐ ฬ+2๐ ฬโ๐ ฬ & perpendicular to ๐ โ = 3๐ ฬ โ 3๐ ฬ is
[๐ โโ(๐ ฬ+๐๐ ฬโ๐ ฬ)]. (๐๐ ฬโ๐๐ ฬ) = 0
Finding Cartesian form
Putting ๐ โ = x๐ ฬ + y๐ ฬ + z๐ ฬ
[(๐ฅ๐ ฬ+๐ฆ๐ ฬ+๐ง๐ ฬ )โ(๐ ฬ+2๐ ฬโ๐ ฬ)]. (3๐ ฬ โ 3๐ ฬ) = 0
[(๐ฅโ1) ๐ ฬ+(๐ฆโ2) ๐ ฬ+ (๐งโ(โ 1))๐ ฬ ]. (3๐ ฬ โ 3๐ ฬ) = 0
3(x โ 1) + 0 (y โ 2) + (โ3)(z + 1) = 0
3x โ 3 โ 3z โ 3 = 0
3x โ 3z โ 6 = 0
3(x โ z โ 2) = 0
x โ z โ 2 = 0
Made by
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.