The equation of the plane passing through the line of intersection of the planes x+y+z=1 and 3 x+4 y+5 z=2 and perpendicular to the XY- plane is

Step-by-step solution

A plane passing through the line of intersection of x+y+z-1=0 and 3x+4y+5z-2=0 has the form (x+y+z-1) + (3x+4y+5z-2) = 0 . Simplifying gives (1+3 )x + (1+4 )y + (1+5 )z - (1+2 ) = 0 with normal vector n = (1+3 ) i + (1+4 ) j + (1+5 ) k . Since this plane is perpendicular to the XY-plane z=0 with normal k , their dot product must vanish: n k = 1+5 = 0 . Solving yields = - 1 5 . Substituting gives 2 5 x + 1 5 y - 3 5 = 0 , which after multiplying by 5 becomes 2x + y - 3 = 0 . The final answer is A .