Find the equations oof two planes through the points (4,2,1) and (2,1,-1) and making of 45 degree with the plane - Maths - Vector Algebra

Question

Find the equations oof two planes through the points (4,2,1) and
(2,1,-1) and making of 45 degree with the plane x-4y+z-9=0

Neha Sethi · Accepted Answer

Dear student
We know that general equation of a  plane through x1,y1,z1 isax-x1+by-y1+cz-z1=0Where a,b,c are the direction ratios of normal to plane
Now, the equation of the plane passing through 4,2,1 beax-4+by-2+cz-1=0  
1Since the plane 1 also passes through 2,1,-1 then it must satisfy 1So, put x=2, y=1and  z=-1 in 1, we geta2-4+b1-2+c-1-1=0-2a-b-2c=0b=-2a-2c  
1We know that the angle θ between the two planes is,cosθ=aa1+bb1+cc1a2+b2+c2a12+b12+c12Now the plane 1, makes an angle of π4 with the plane x-4y+z-9=0cosπ4=a×1+b×-4+c×1a2+b2+c212+-42+1212=a-4b+ca2+b2+c21+16+112=a-4b+ca2+b2+c2323=a-4b+ca2+b2+c23a2+b2+c2=a-4b+c9a2+9b2+9c2=a2+16b2+c2-8ab-8bc+2ac8a2-7-2a-2c2+8c2=2ac-4a-2a-2c-4-2a-2cc8a2-74a2+4c2+8ac+8c2=2ac+8a2+8ac+8ac+8c2   using 18a2-28a2-28c2-56ac+8c2=16a2+16c2+34ac-36a2-36c2-56ac-34ac=0-36a2-36c2-90ac=012a2+12c2+30ac=0Please recheck it cannot be solved further,
Regards

Find the equations oof two planes through the points (4,2,1) and (2,1,-1) and making of 45 degree with the plane x-4y+z-9=0