The origin is shifted to the point (2,3) by translation of axes and then the coordinate axes are rotated about the origin through an angle in the counter-clockwise sense. Due to this if the equation 3 x^2+2 x y+3 y^2-18 x-22 y+50=0 is transformed to 4 x^2+2 y^2-1=0 , then the angle =

Step-by-step solution

3 x^2+2 x y+3 y^2-18 x-22 y+50=0 Shifting origin to the point (2,3) , put x=X+2, y=Y+3 aligned & 3(X+2)^2+2(X+2)(Y+3)+3(Y+3)^2 & -18(X+2)-22(Y+3)+50=0 & 3 X^2+2 X Y+3 Y^2-1=0 aligned Now, for rotating about angle, put aligned & X=x^ -y^ and Y=x^ +y^ & 3 (x^ -y^ )^2+2 (x^ -y^ ) & (x^ +y^ )+3 (x^ +y^ )^2-1=0 & (3 ^2 +3 ^2 +2 ) x^ 2 & + (3 ^2 +3 ^2 -2 ) y¹² & + (2 ^2 -2 ^2 -6 . & +6 ) x^ y^ -1=0 & (3+ 2 ) x^ 2 +(3- 2 ) y^ 2 +(2 2 ) x^ y^ -1=0 aligned Comparing with 4 x^2+2 y^2-1=0 we get, 2 2 =0 = 4