When the coordinate axes are rotated about the origin in the positive direction through an angle ( 4 ), if the equation (25 x^2+9 y^2=225 ) is transformed to ( x^2+ x y+ y^2= ), then (( + + - )^2= )

Step-by-step solution

After rotation of coordinate axes about the origin in the positive direction through on angle ( 4 ), the new coordinates are ((X, Y) ) have relation with older coordinates ((x, y) ) is ( aligned & (x, y)=[(X -Y ),(Y +X )) , where & = 4 & = ( ( X 2 - Y 2 ), ( Y 2 + X 2 ) ) aligned ) so, (25 x^2+9 y^2=225 ) becomes ( aligned & 25 ( X-Y 2 )^2+9 ( X+Y 2 )^2=225 & 34 X^2+34 Y^2-32 X Y=450 & 17 X^2+17 Y^2-16 X Y=225 aligned ) On comparing, we get ( aligned & = =17, =-16 and =225 & ( + + - )^2 & =(34-16-15)^2 & =3^2=9 ali