Question

Find the equation of plane passing through the line of intersection of the planes $4x-5y-4z=1$ and $2x+y+2z=8$ and the point $(2,1,3)$.

Answer 1

Let the equation of the plane be

$(2x+y+2z-8)+a(4x-5y-4z-1)=0$

On satisfying with $(2,1,3)$ we get

$(4+1+6-8)+a(8-5-12-1)=0$

$⟹a=3/10$

So, equation will be

$(2x+y+2z-8)+\frac{3}{10}(4x-5y-4z-1)=0$

$⟹32x-5y+9z=83$