In order to eliminate the first degree terms from the equation $4x^2+8xy+10y^2-8x-44y+14=0$,the point to which the origin has to be shifted is

$A$ 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

The point to which the origin should be shifted so that the equation $y^2-6y-4x+13=0$ is transformed to the form $y^2+Ax=0$ is

$A$ vector $\vec{a}$ has components $3p$ and $1$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If,with respect to the new system,$\vec{a}$ has components $p+1$ and $\sqrt{10}$,then a value of $p$ is equal to

When the equation $x^2 - 3xy + 11x - 12y + 36 = 0$ is transformed by shifting the origin to the point $(-4, 1)$ while keeping the axes parallel,it becomes $ax^2 + bxy + 1 = 0$. Then $b^2 - a = \dots$

If the coordinate axes are rotated through an angle $\frac{\pi}{6}$ about the origin,then the transformed equation of $\sqrt{3} x^2-4 x y+\sqrt{3} y^2=0$ is