The point to which the origin should be shifted so that the equation $y^2-6y-4x+13=0$ will not contain any term in $y$ and the constant term,is

Point $(-1, 2)$ is changed to $(a, b)$ when the origin is shifted to the point $(2, -1)$ by translation of axes. Point $(a, b)$ is changed to $(c, d)$ when the axes are rotated through an angle of $45^{\circ}$ about the new origin. Point $(c, d)$ is changed to $(e, f)$ when $(c, d)$ is reflected through the line $y = x$. Then $(e, f) =$

$A$ 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

Suppose the axes are to be rotated through an angle $\theta$ so as to remove the $xy$ term from the equation $3x^2+2\sqrt{3}xy+y^2=0$. Then in the new coordinate system,the equation $x^2+y^2+2xy=2$ is transformed to:

When the coordinate axes are rotated about the origin in the positive direction through an angle $\frac{\pi}{4}$,if the equation $25x^2+9y^2=225$ is transformed to $\alpha x^2+\beta xy+\gamma y^2=\delta$,then $(\alpha+\beta+\gamma-\sqrt{\delta})^2=$

The transformed equation of $3x^2 - 4xy = r^2$ when the coordinate axes are rotated through an angle $\tan^{-1}(2)$ is: