Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If λ1<λ2 are two values of λ such that the angle between the planes P1:r(3ˆi5ˆj+ˆk)=7 and P2:r(λˆi+ˆj3ˆk)=9 is sin1(265), then the square of the length of perpendicular from the point (38λ1,10λ2,2) to the plane P1 is ______

JEE MainJEE Main 2023Three Dimensional Geometry

Solution:

P1=r(3ˆi5ˆj+ˆk)=7
P2=r(λˆi+ˆj3ˆk)=9
θ=sin1(265)
sinθ=265
cosθ=15.
cosθ=rr|r||r2|
=(3i5j+K)(λi+j3K)35λ2+10
15=|3λ835λ2+10|
 Square 125=9λ2+6448λ35(λ2+10)
19λ2120λ+125=0
19λ295λ25λ+125=0
x=5,2519
image
Perpendicular distance of point
(38λ1,10λ2,2)=(50,50,2) from plane P1
=|3×505×50+27|35=10535
Square =105×10535=315