If the axes are transformed to the point $(-1, 1)$,then the equation $3x^2 + y^2 + 2x + 4y + 15 = 0$ would transform to:

Let $P$ be the point to which the origin is shifted by the translation of axes so as to remove the first-degree terms from the equation $3x^2+y^2-6x+4y+4=0$. If the origin is shifted to $P$ by the translation of axes,then the transformed equation of $2x^2+3xy-5y^2+2x-23y-24=0$ is

If the equation of a curve $C$ is transformed to $9x^2 + 25y^2 = 225$ by the rotation of the coordinate axes about the origin through an angle $\frac{\pi}{4}$ in the positive direction,then the equation of the curve $C$ before the transformation is:

By shifting the origin to the point $(-1, 2)$ through translation of axes,if $ax^2+2hxy+by^2+2gx+2fy+c=0$ is the transformed equation of $2x^2-xy+y^2-3x+4y-5=0$,then $2(f+g+h)=$

If the equation $3x^2 + 4y^2 - xy + k = 0$ is the transformed equation of $3x^2 + 4y^2 - xy - 5x - 7y + 2 = 0$ after shifting the origin to the point $(\alpha, \beta)$ by the translation of axes,then $\alpha + \beta - k =$

$A$ line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a given angle $\theta$ keeping the origin fixed,this line $L$ has the intercepts $p$ and $q$. Then