The equation of a curve referred to a given system of axes is `3x^2+2x y+3y^2=10.` Find its equation if the axes are rotated through an angle `45^0` , the origin remaining unchanged.

Explanation

We have
`x=Xcostheta-Ysin theta`
`=X cos theta 45^@-Ysin45^@=(X)/(sqrt2)+(Y)/(sqrt2)`
`y=X sintheta+Ycostheta`
`=X sintheta 45^@-Ycos45^@=(X)/(sqrt2)+(Y)/(sqrt2)`
Hence, the equation `3x^2+2xy+3y^2=10` transforms to
`3((X)/(sqrt2)-(Y)/(sqrt2))^2+2((X)/(sqrt2)-(Y)/(sqrt2))((X)/(sqrt2)+(Y)/(sqrt2))+3((X)/(sqrt2)+(Y)/(sqrt2))^2=10`
or `2X^2+Y^2=5`