The equation of the plane passing through the intersection of planes 2x - y + z = 3 , 4x - 3y + 5z = -9 and parallel to the line x+1 2 = y+3 4 = z-3 5 is...

Step-by-step solution

The equation of the plane passing through the intersection of the given planes is (2x - y + z - 3) + (4x - 3y + 5z + 9) = 0 (2 + 4 )x - (1 + 3 )y + (1 + 5 )z - (3 - 9 ) = 0 Since this plane is parallel to the line x+1 2 = y+3 4 = z-3 5 , its normal is perpendicular to the line. Therefore, 2(2 + 4 ) - 4(1 + 3 ) + 5(1 + 5 ) = 0 4 + 8 - 4 - 12 + 5 + 25 = 0 21 + 5 = 0 = - 5 21 Substituting = - 5 21 in the equation of the plane, we get (2x - y + z - 3) - 5 21 (4x - 3y + 5z + 9) = 0 21(2x - y + z - 3) - 5(4x - 3y + 5z +