The direction ratios of a normal to the plane through (1, 0, 0) and (0, 1, 0), which makes an angle of π4 with the plane x + y = 3, are

Any plane through ( 1, 0, 0) is a(x−1)+by+cz=0----iIt passes through (0, 1, 0). Therefore, a(0−1)+b(1)+c(0)=0 or −a+b=0----ii(i) makes an angle of π4 with x + y=3,thereforecos⁡π4=a(1)+b(1)+c(0)a2+b2+c21+1+0or 12=a+b2a2+b2+c2or a+b=a2+b2+c2Squaring, we get a2+b2+2ab=a2+b2+c2or 2ab=c2 or 2a2=c2or c=2aHence a:b:c=a:a:2a=1:1:2