Given, transformed equation is x2+y2−6x+8y+21=0 Now,X′=xcosπ∕4−Ysinπ∕4=
X−Y
√2
&Y′=xsinπ∕4+Ycosπ∕4=
X+Y
2
Before transformation, the equation is (
X−Y
√2
)2+(
X+Y
√2
)2−6(
X−Y
√2
)+8(
X+Y
√2
) ⇒X2+Y2−2XY+X2+Y2+2XY −6√2X+6√2Y+8√2X+8√2Y+42=0 =2x2+2y2+2√2x+14√2y+42=0 ⇒X2+Y2+√2X+7√2Y+21=0 comparing this to, aX2+bY2+cX+dY+e=0, we get a=1,b=1,c=√2,d=7√2,e=21 ∴(a+b+c2+d2−5e)2 =(1+1+2+98−105)2 =(102−105)2 =9