The equation of plane passing through (1,0,0) and (0,1,0) and making an angle 45^ with the plane x+y-3=0 is

Step-by-step solution

The required plane passes through (1, 0, 0) and (0, 1, 0) , and makes a 45^ angle with the plane x + y - 3 = 0 . Let the general equation be Ax + By + Cz + D = 0 . Substituting (1, 0, 0) gives A + D = 0 A = -D . Substituting (0, 1, 0) gives B + D = 0 B = -D . Using these, the equation becomes -Dx - Dy + Cz + D = 0 . For D 0 , divide by D : -x - y + C D z + 1 = 0 . Let k = C D , so the plane becomes x + y - kz - 1 = 0 . The normal vector is n ₁ = (1, 1, -k) . The given plane x + y - 3 = 0 has normal vector n ₂ = (1,