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Question

If the axes are shifted to (2,3) and then rotated through π4 in anticlockwise direction, then transformed equation of x2y2+2x+4y=0 is

A
8x+12y22xy212=0
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B
8x12y22xy212=0
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C
8x+12y+22xy212=0
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D
8x+12y+22xy+212=0
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Solution

The correct option is A 8x+12y22xy212=0
As (0,0) is shifted to (2,3)
x=X2,y=Y3
Where (x,y) are the original coordinates and (X,Y) are transformed coordinates.
After origin shifting axes are rotated through π4
X=Xcosπ4Ysinπ4=XY2X=Xcosπ4+Ysinπ4=X+Y2
Where (X,Y) are the rotated coordinates,
Therefore,
x=X2=XY22y=Y3=X+Y23
Now, finding the transformed equation for
x2y2+2x+4y=0
Putting the respective values of (x,y), we get
(XY22)2(X+Y23)2+2(XY22)+4(X+Y23)=022(XY)+102(X+Y)2XY21=08X+12Y22XY212=0
So, the transformed equation is
8x+12y22xy212=0

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