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 $b^2 - a = \dots$

The mixed term $xy$ is to be removed from the general equation $ax^2 + by^2 + 2hxy + 2fy + 2gx + c = 0$. One should rotate the axes through an angle $\theta$ given by $\tan 2\theta$ equal to:

By rotating the axes through an angle of $30^{\circ}$ in the anti-clockwise direction about the origin,the equation $4x^2+12xy+9y^2+6x+9y+2=0$ becomes $ax^2+2hxy+by^2+2gx+2fy+c=0$. Then which of the following is true?

The point $P(\alpha, \beta)$ with $\alpha > 0, \beta > 0$ undergoes the following transformations successively:
$a)$ Translation by $3$ units in the positive direction of the $x$-axis.
$b)$ Reflection about the line $y = -x$.
$c)$ Rotation of axes through an angle of $\frac{\pi}{4}$ about the origin in the positive direction.
If the final position of the point $P$ is $(-4\sqrt{2}, -2\sqrt{2})$,then find the value of $(\alpha + \beta)$.

The angle by which axes are to be rotated without changing the origin so that the transformed equation of $x^2+4xy-y^2=0$ in new coordinates $(X, Y)$ does not contain the $XY$ term is

When the coordinate axes are rotated by an angle $\tan^{-1}\left(\frac{3}{4}\right)$ about the origin,then the equation $x^2+y^2=9$ is transformed to the equation