Transformation of Axes

The angle through which the axes may be turned, so that the equation $Ax+By+C=0\left(A\ne 0\right)$ may be reduced to the form $x=$constant, is

The axes be shifted without rotation, so that the equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ does not contain terms in $x, y$ and constant term, then the point is

If by rotating the coordinate axes without translating the origin, the expression $a_1{x}^2+2h_{1}xy+b_1{y}^2$ becomes $a_2{X}^2+2h_{2}XY+b_2{Y}^2,$ then

If by rotating the coordinate axes without translating the origin, the expression $a_1{x}^2+2h_{1}xy+b_1{y}^2$ becomes $a_2{X}^2+2h_{2}XY+b_2{Y}^2,$ then

If by rotating the coordinate axes without translating the origin, the expression $a_1{x}^2+2h_{1}xy+b_1{y}^2$ becomes $a_2{X}^2+2h_{2}XY+b_2{Y}^2,$ then

The angle through which the coordinates axes be rotated so that $xy$ term in the equation $5x^2+4\sqrt{3}xy+9y^2=0$ may be missing, is

If $(x,y)$ and $(X,Y)$ are the coordinates of the same point referred to two sets of rectangular axes with the same origin and if $ax+by$ becomes $pX+qY,$ where $a, b$ are independent of $x, y,$ then

If the axes are rotated through an angle of $45°$ in the clockwise direction, the coordinates of a point in the new system are $(0,-2),$ then its original coordinates

Shift the origin to a suitable point so that the equation $y^2+4y+8x-2=0$ will not contain term in $y$ and the constant term, then the origin is

If the coordinates of a point $(4,5)$ become $(-3,9)$, then the point of origin be shifted at

If in a $\mathrm{\Delta }ABC$ (whose circumcentre is origin), $a\le \sin A,$ then for any point $(x,y)$ inside the circumcircle of $\mathrm{\Delta }ABC$, then