Question

Determine the vector and cartesian equations of the plane that passes through the line of intersection of the planes r⃗ .(2i^+2j^–3k^)=7 and r⃗ .(2i^+5j^+3k^)=9 such that the intercept made by the plane on x-axis and z-axis should be equal.

Answer 1

Answer:

Given Planes (Cartesian form)

2x+2y−3z−7=0

and 2x+3y+3z−9=0

Plane passing through their point of intersection

⇒(2x+2y−3z−7)+λ(2x++5y+3z−9)=0

⇒x(2+2λ)+y(2+5λ)+z(−3+3λ)+(−9λ−7)=0

⇒x(9λ+72+2λ )+y( 9λ+72+5λ )+z( 9λ+73λ−3 )=1

∵x and z intercepts are equal

⇒9λ+72+2λ

= 9λ+73λ−3

⇒2+2λ=3λ−3

⇒λ=5

∴ Required equation of plane

⇒x( 5212 )+y( 5227)+z( 5212 )=1

⇒12x+27y+12z=52

⇒4x+9y+4z=14

Vector form

⇒ r .(4i +9 j +4 k )−14=0

hope it helps you ♥️♥️

with regards @ARSH☺️☺️