The correct option is A 8x+12y−2√2xy−21√2=0
As (0,0) is shifted to (−2,−3)
x=X−2,y=Y−3
Where (x,y) are the original coordinates and (X,Y) are transformed coordinates.
∵ After origin shifting axes are rotated through π4
X=X′cosπ4−Y′sinπ4=X′−Y′√2X=X′cosπ4+Y′sinπ4=X′+Y′√2
Where (X′,Y′) are the rotated coordinates,
Therefore,
x=X−2=X′−Y′√2−2y=Y−3=X′+Y′√2−3
Now, finding the transformed equation for
x2−y2+2x+4y=0
Putting the respective values of (x,y), we get
(X′−Y′√2−2)2−(X′+Y′√2−3)2+2(X′−Y′√2−2)+4(X′+Y′√2−3)=0⇒−2√2(X′−Y′)+10√2(X′+Y′)−2X′Y′−21=0⇒8X′+12Y′−2√2X′Y′−21√2=0
So, the transformed equation is
8x+12y−2√2xy−21√2=0