To which point should the origin be shifted in order to eliminate the first-degree terms ($x$ and $y$ terms) from the equation $4x^2 + 9y^2 - 8x + 36y + 4 = 0$?

If a square $ABCD$,where $A(0,0), B(2,0), C(2,2)$ and $D(0,2)$ undergoes the following transformations successively,then the final figure would be a:
$(i)$ $f_1(x, y) \longrightarrow (y, x)$
(ii) $f_2(x, y) \longrightarrow (x+3y, y)$
(iii) $f_3(x, y) \longrightarrow \left(\frac{x-y}{2}, \frac{x+y}{2}\right)$

Two fixed points are $A(a, 0)$ and $B(-a, 0)$. If $\angle A - \angle B = \theta$,then the locus of point $C$ of triangle $ABC$ will be

If the polar coordinates of a point are $\left( \sqrt{2}, -\frac{3\pi}{4} \right)$,what are its Cartesian coordinates?

If the Cartesian coordinates of a point are $\left(\frac{-5 \sqrt{3}}{2}, \frac{5}{2}\right)$,then its polar coordinates are

The points $(-a, -b)$,$(0, 0)$,$(a, b)$,and $(a^2, ab)$ are