The equation of the plane passing through the point of intersection of the planes 2 x-y+z-3=0 and 4 x-3 y+5 z+9=0 and parallel to the lines x+1 2 = y+3 4 = z-3 5 is x+ y+ z+d=0 Then + + + d =

Step-by-step solution

Plane through intersection: P₁ + P₂ = (2x - y + z - 3) + (4x - 3y + 5z + 9) = 0 Rewriting: (2 + 4 )x + (-1 - 3 )y + (1 + 5 )z + (-3 + 9 ) = 0 The plane is parallel to the line with direction vector v = 2 i + 4 j + 5 k . The normal vector n = (2 + 4 ) i + (-1 - 3 ) j + (1 + 5 ) k must be perpendicular to v , so their dot product is zero: n v = (2 + 4 )(2) + (-1 - 3 )(4) + (1 + 5 )(5) = 0 Simplifying: 4 + 8 - 4 - 12 + 5 + 25 = 0 5 + 21 = 0 = - 5 21 Substitute back into the plane equation: (2 + 4(- 5 21 ))x + (-1 - 3(